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  <title>The <text font="typewriter">physics</text> package</title>
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    <tags>
      <tag>1</tag>
      <tag role="refnum">1</tag>
      <tag role="typerefnum">§1</tag>
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    <title><tag close=" ">1</tag>List of commands</title>
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      <tags>
        <tag>1.1</tag>
        <tag role="refnum">1.1</tag>
        <tag role="typerefnum">§1.1</tag>
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      <title><tag close=" ">1.1</tag>Automatic bracing</title>
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                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex5.m1.20">
                    <XMApp>
                      <XMTok meaning="evaluated-at"/>
                      <XMRef idref="S1.Ex5.m1.13"/>
                      <XMRef idref="S1.Ex5.m1.14"/>
                      <XMRef idref="S1.Ex5.m1.15"/>
                    </XMApp>
                    <XMApp>
                      <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                      <XMApp>
                        <XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="false">[</XMTok>
                          <XMApp scriptpos="mid" xml:id="S1.Ex5.m1.13">
                            <XMTok role="SUPERSCRIPTOP" scriptpos="mid1"/>
                            <XMApp scriptpos="mid">
                              <XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
                              <XMTok mathstyle="display" meaning="sum" role="SUMOP" scriptpos="mid">∑</XMTok>
                              <XMApp>
                                <XMTok fontsize="70%" meaning="equals" role="RELOP">=</XMTok>
                                <XMTok font="italic" fontsize="70%" role="UNKNOWN">X</XMTok>
                                <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                              </XMApp>
                            </XMApp>
                            <XMTok font="italic" fontsize="70%" role="UNKNOWN">N</XMTok>
                          </XMApp>
                          <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                        </XMWrap>
                        <XMTok fontsize="70%" meaning="0" role="NUMBER" xml:id="S1.Ex5.m1.14">0</XMTok>
                      </XMApp>
                      <XMTok fontsize="70%" meaning="infinity" name="infty" role="ID" xml:id="S1.Ex5.m1.15">∞</XMTok>
                    </XMApp>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex6">
          <Math mode="display" tex="\order{x^{2}}\qquad\order\Big{x^{2}}\qquad\order*{\frac{X}{Y}}\qquad%&#10;\commutator{A}{B}\qquad\commutator\Big{A}{B}\qquad\commutator*{A}{\frac{X}{Y}}%&#10;\qquad\anticommutator{A}{B}\qquad\poissonbracket{A}{B}" text="list@(order@(x ^ 2), order@(x ^ 2), order@(X / Y), commutator@(A, B), commutator@(A, B), commutator@(A, X / Y), anticommutator@(A, B), poisson-bracket@(A, B))" xml:id="S1.Ex6.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex6.m1.14"/>
                  <XMRef idref="S1.Ex6.m1.15"/>
                  <XMRef idref="S1.Ex6.m1.16"/>
                  <XMRef idref="S1.Ex6.m1.17"/>
                  <XMRef idref="S1.Ex6.m1.18"/>
                  <XMRef idref="S1.Ex6.m1.19"/>
                  <XMRef idref="S1.Ex6.m1.20"/>
                  <XMRef idref="S1.Ex6.m1.21"/>
                </XMApp>
                <XMWrap>
                  <XMDual xml:id="S1.Ex6.m1.14">
                    <XMApp>
                      <XMTok meaning="order"/>
                      <XMRef idref="S1.Ex6.m1.1"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">(</XMTok>
                      <XMApp xml:id="S1.Ex6.m1.1">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                        <XMTok font="italic" role="UNKNOWN">x</XMTok>
                        <XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
                      </XMApp>
                      <XMTok role="CLOSE" stretchy="true">)</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex6.m1.15">
                    <XMApp>
                      <XMTok meaning="order"/>
                      <XMRef idref="S1.Ex6.m1.2"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok fontsize="160%" role="OPEN" stretchy="false">(</XMTok>
                      <XMApp xml:id="S1.Ex6.m1.2">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                        <XMTok font="italic" role="UNKNOWN">x</XMTok>
                        <XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
                      </XMApp>
                      <XMTok fontsize="160%" role="CLOSE" stretchy="false">)</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex6.m1.16">
                    <XMApp>
                      <XMTok meaning="order"/>
                      <XMRef idref="S1.Ex6.m1.3"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="false">(</XMTok>
                      <XMApp xml:id="S1.Ex6.m1.3">
                        <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok role="CLOSE" stretchy="false">)</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex6.m1.17">
                    <XMApp>
                      <XMTok meaning="commutator"/>
                      <XMRef idref="S1.Ex6.m1.4"/>
                      <XMRef idref="S1.Ex6.m1.5"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">[</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex6.m1.4">A</XMTok>
                      <XMTok role="PUNCT">,</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex6.m1.5">B</XMTok>
                      <XMTok role="CLOSE" stretchy="true">]</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex6.m1.18">
                    <XMApp>
                      <XMTok meaning="commutator"/>
                      <XMRef idref="S1.Ex6.m1.6"/>
                      <XMRef idref="S1.Ex6.m1.7"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok fontsize="160%" role="OPEN" stretchy="false">[</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex6.m1.6">A</XMTok>
                      <XMTok role="PUNCT">,</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex6.m1.7">B</XMTok>
                      <XMTok fontsize="160%" role="CLOSE" stretchy="false">]</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex6.m1.19">
                    <XMApp>
                      <XMTok meaning="commutator"/>
                      <XMRef idref="S1.Ex6.m1.8"/>
                      <XMRef idref="S1.Ex6.m1.9"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="false">[</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex6.m1.8">A</XMTok>
                      <XMTok role="PUNCT">,</XMTok>
                      <XMApp xml:id="S1.Ex6.m1.9">
                        <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok role="CLOSE" stretchy="false">]</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex6.m1.20">
                    <XMApp>
                      <XMTok meaning="anticommutator"/>
                      <XMRef idref="S1.Ex6.m1.10"/>
                      <XMRef idref="S1.Ex6.m1.11"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">{</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex6.m1.10">A</XMTok>
                      <XMTok role="PUNCT">,</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex6.m1.11">B</XMTok>
                      <XMTok role="CLOSE" stretchy="true">}</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex6.m1.21">
                    <XMApp>
                      <XMTok meaning="poisson-bracket"/>
                      <XMRef idref="S1.Ex6.m1.12"/>
                      <XMRef idref="S1.Ex6.m1.13"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">{</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex6.m1.12">A</XMTok>
                      <XMTok role="PUNCT">,</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex6.m1.13">B</XMTok>
                      <XMTok role="CLOSE" stretchy="true">}</XMTok>
                    </XMWrap>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
      </para>
    </subsection>
    <subsection inlist="toc" xml:id="S1.SS2">
      <tags>
        <tag>1.2</tag>
        <tag role="refnum">1.2</tag>
        <tag role="typerefnum">§1.2</tag>
      </tags>
      <title><tag close=" ">1.2</tag>Vector notation</title>
      <para xml:id="S1.SS2.p1">
        <equation xml:id="S1.Ex7">
          <Math mode="display" tex="\mathbf{a}\qquad{\bf\it a}\qquad\overrightarrow{\mathbf{a}}\qquad%&#10;\overrightarrow{{\bf\it a}}\qquad\hat{\mathbf{a}}\qquad\hat{{\bf\it a}}" text="list@(a, a, overrightarrow@(a), overrightarrow@(a), hat@(a), hat@(a))" xml:id="S1.Ex7.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex7.m1.1"/>
                  <XMRef idref="S1.Ex7.m1.2"/>
                  <XMRef idref="S1.Ex7.m1.3"/>
                  <XMRef idref="S1.Ex7.m1.4"/>
                  <XMRef idref="S1.Ex7.m1.5"/>
                  <XMRef idref="S1.Ex7.m1.6"/>
                </XMApp>
                <XMWrap>
                  <XMTok font="bold" role="ID" xml:id="S1.Ex7.m1.1">a</XMTok>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMTok font="bold italic" role="ID" xml:id="S1.Ex7.m1.2">a</XMTok>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMApp role="ID" xml:id="S1.Ex7.m1.3">
                    <XMTok name="overrightarrow" role="OVERACCENT">→</XMTok>
                    <XMTok font="bold" role="UNKNOWN">a</XMTok>
                  </XMApp>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMApp role="ID" xml:id="S1.Ex7.m1.4">
                    <XMTok name="overrightarrow" role="OVERACCENT">→</XMTok>
                    <XMTok font="bold italic" role="UNKNOWN">a</XMTok>
                  </XMApp>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMApp role="ID" xml:id="S1.Ex7.m1.5">
                    <XMTok name="hat" role="OVERACCENT" stretchy="false">^</XMTok>
                    <XMTok font="bold" role="UNKNOWN">a</XMTok>
                  </XMApp>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMApp role="ID" xml:id="S1.Ex7.m1.6">
                    <XMTok name="hat" role="OVERACCENT" stretchy="false">^</XMTok>
                    <XMTok font="bold italic" role="UNKNOWN">a</XMTok>
                  </XMApp>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex8">
          <Math mode="display" tex="\dotproduct\qquad\crossproduct\qquad\crossproduct" text="list@(dot-product, cross-product, cross-product)" xml:id="S1.Ex8.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex8.m1.1"/>
                  <XMRef idref="S1.Ex8.m1.2"/>
                  <XMRef idref="S1.Ex8.m1.3"/>
                </XMApp>
                <XMWrap>
                  <XMTok meaning="dot-product" name="dotproduct" role="MULOP" xml:id="S1.Ex8.m1.1">⋅</XMTok>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMTok meaning="cross-product" name="crossproduct" role="MULOP" xml:id="S1.Ex8.m1.2">×</XMTok>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMTok meaning="cross-product" name="crossproduct" role="MULOP" xml:id="S1.Ex8.m1.3">×</XMTok>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex9">
          <Math mode="display" tex="\gradient\qquad\gradient{\Psi}\qquad\gradient(\Psi+X^{Y})\qquad\gradient[\Psi+%&#10;X^{Y}]" text="list@(gradient, gradient@(Psi), gradient@(Psi + X ^ Y), gradient@(Psi + X ^ Y))" xml:id="S1.Ex9.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex9.m1.6"/>
                  <XMRef idref="S1.Ex9.m1.7"/>
                  <XMRef idref="S1.Ex9.m1.8"/>
                  <XMRef idref="S1.Ex9.m1.9"/>
                </XMApp>
                <XMWrap>
                  <XMTok meaning="gradient" name="nabla" role="OPERATOR" xml:id="S1.Ex9.m1.6">∇</XMTok>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMApp xml:id="S1.Ex9.m1.7">
                    <XMTok meaning="gradient" name="nabla" role="OPERATOR">∇</XMTok>
                    <XMTok name="Psi" role="UNKNOWN" xml:id="S1.Ex9.m1.1">Ψ</XMTok>
                  </XMApp>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex9.m1.8">
                    <XMApp>
                      <XMTok meaning="gradient"/>
                      <XMRef idref="S1.Ex9.m1.2"/>
                    </XMApp>
                    <XMApp>
                      <XMTok name="nabla" role="OPERATOR" xml:id="S1.Ex9.m1.4">∇</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex9.m1.2">
                          <XMTok meaning="plus" role="ADDOP">+</XMTok>
                          <XMTok name="Psi" role="UNKNOWN">Ψ</XMTok>
                          <XMApp>
                            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                            <XMTok font="italic" role="UNKNOWN">X</XMTok>
                            <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                          </XMApp>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex9.m1.9">
                    <XMApp>
                      <XMTok meaning="gradient"/>
                      <XMRef idref="S1.Ex9.m1.3"/>
                    </XMApp>
                    <XMApp>
                      <XMTok name="nabla" role="OPERATOR" xml:id="S1.Ex9.m1.5">∇</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">[</XMTok>
                        <XMApp xml:id="S1.Ex9.m1.3">
                          <XMTok meaning="plus" role="ADDOP">+</XMTok>
                          <XMTok name="Psi" role="UNKNOWN">Ψ</XMTok>
                          <XMApp>
                            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                            <XMTok font="italic" role="UNKNOWN">X</XMTok>
                            <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                          </XMApp>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">]</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex10">
          <Math mode="display" tex="\divergence\qquad\divergence{\mathbf{a}}\qquad\divergence(\mathbf{a}+X^{Y})%&#10;\qquad\divergence[\mathbf{a}+X^{Y}]" text="list@(divergence, divergence@(a), divergence@(a + X ^ Y), divergence@(a + X ^ Y))" xml:id="S1.Ex10.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex10.m1.4"/>
                  <XMRef idref="S1.Ex10.m1.5"/>
                  <XMRef idref="S1.Ex10.m1.6"/>
                  <XMRef idref="S1.Ex10.m1.7"/>
                </XMApp>
                <XMWrap>
                  <XMDual role="OPERATOR" xml:id="S1.Ex10.m1.4">
                    <XMTok meaning="divergence"/>
                    <XMWrap>
                      <XMTok name="nabla" role="OPERATOR">∇</XMTok>
                      <XMTok name="cdot" role="MULOP">⋅</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex10.m1.5">
                    <XMApp>
                      <XMTok meaning="divergence"/>
                      <XMRef idref="S1.Ex10.m1.1"/>
                    </XMApp>
                    <XMApp>
                      <XMTok name="cdot" role="MULOP">⋅</XMTok>
                      <XMTok name="nabla" role="OPERATOR">∇</XMTok>
                      <XMTok font="bold" role="ID" xml:id="S1.Ex10.m1.1">a</XMTok>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex10.m1.6">
                    <XMApp>
                      <XMTok meaning="divergence"/>
                      <XMRef idref="S1.Ex10.m1.2"/>
                    </XMApp>
                    <XMApp>
                      <XMTok name="cdot" role="MULOP">⋅</XMTok>
                      <XMTok name="nabla" role="OPERATOR">∇</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex10.m1.2">
                          <XMTok meaning="plus" role="ADDOP">+</XMTok>
                          <XMTok font="bold" role="ID">a</XMTok>
                          <XMApp>
                            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                            <XMTok font="italic" role="UNKNOWN">X</XMTok>
                            <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                          </XMApp>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex10.m1.7">
                    <XMApp>
                      <XMTok meaning="divergence"/>
                      <XMRef idref="S1.Ex10.m1.3"/>
                    </XMApp>
                    <XMApp>
                      <XMTok name="cdot" role="MULOP">⋅</XMTok>
                      <XMTok name="nabla" role="OPERATOR">∇</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">[</XMTok>
                        <XMApp xml:id="S1.Ex10.m1.3">
                          <XMTok meaning="plus" role="ADDOP">+</XMTok>
                          <XMTok font="bold" role="ID">a</XMTok>
                          <XMApp>
                            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                            <XMTok font="italic" role="UNKNOWN">X</XMTok>
                            <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                          </XMApp>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">]</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex11">
          <Math mode="display" tex="\curl\qquad\curl{\mathbf{a}}\qquad\curl(\mathbf{a}+X^{Y})\qquad\curl[\mathbf{a%&#10;}+X^{Y}]" text="list@(curl, curl@(a), curl@(a + X ^ Y), curl@(a + X ^ Y))" xml:id="S1.Ex11.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex11.m1.4"/>
                  <XMRef idref="S1.Ex11.m1.5"/>
                  <XMRef idref="S1.Ex11.m1.6"/>
                  <XMRef idref="S1.Ex11.m1.7"/>
                </XMApp>
                <XMWrap>
                  <XMDual role="OPERATOR" xml:id="S1.Ex11.m1.4">
                    <XMTok meaning="curl"/>
                    <XMWrap>
                      <XMTok name="nabla" role="OPERATOR">∇</XMTok>
                      <XMTok meaning="cross-product" name="crossproduct" role="MULOP">×</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex11.m1.5">
                    <XMApp>
                      <XMTok meaning="curl"/>
                      <XMRef idref="S1.Ex11.m1.1"/>
                    </XMApp>
                    <XMApp>
                      <XMTok meaning="cross-product" name="crossproduct" role="MULOP">×</XMTok>
                      <XMTok name="nabla" role="OPERATOR">∇</XMTok>
                      <XMTok font="bold" role="ID" xml:id="S1.Ex11.m1.1">a</XMTok>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex11.m1.6">
                    <XMApp>
                      <XMTok meaning="curl"/>
                      <XMRef idref="S1.Ex11.m1.2"/>
                    </XMApp>
                    <XMApp>
                      <XMTok meaning="cross-product" name="crossproduct" role="MULOP">×</XMTok>
                      <XMTok name="nabla" role="OPERATOR">∇</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex11.m1.2">
                          <XMTok meaning="plus" role="ADDOP">+</XMTok>
                          <XMTok font="bold" role="ID">a</XMTok>
                          <XMApp>
                            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                            <XMTok font="italic" role="UNKNOWN">X</XMTok>
                            <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                          </XMApp>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex11.m1.7">
                    <XMApp>
                      <XMTok meaning="curl"/>
                      <XMRef idref="S1.Ex11.m1.3"/>
                    </XMApp>
                    <XMApp>
                      <XMTok meaning="cross-product" name="crossproduct" role="MULOP">×</XMTok>
                      <XMTok name="nabla" role="OPERATOR">∇</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">[</XMTok>
                        <XMApp xml:id="S1.Ex11.m1.3">
                          <XMTok meaning="plus" role="ADDOP">+</XMTok>
                          <XMTok font="bold" role="ID">a</XMTok>
                          <XMApp>
                            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                            <XMTok font="italic" role="UNKNOWN">X</XMTok>
                            <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                          </XMApp>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">]</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex12">
          <Math mode="display" tex="\laplacian\qquad\laplacian{\Psi}\qquad\laplacian(\Psi+X^{Y})\qquad\laplacian[%&#10;\Psi+X^{Y}]" text="list@(laplacian, laplacian@(Psi), laplacian@(Psi + X ^ Y), laplacian@(Psi + X ^ Y))" xml:id="S1.Ex12.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex12.m1.6"/>
                  <XMRef idref="S1.Ex12.m1.7"/>
                  <XMRef idref="S1.Ex12.m1.8"/>
                  <XMRef idref="S1.Ex12.m1.9"/>
                </XMApp>
                <XMWrap>
                  <XMDual role="OPERATOR" xml:id="S1.Ex12.m1.6">
                    <XMTok meaning="laplacian"/>
                    <XMApp>
                      <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                      <XMTok name="nabla" role="OPERATOR">∇</XMTok>
                      <XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMApp xml:id="S1.Ex12.m1.7">
                    <XMApp meaning="laplacian">
                      <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                      <XMTok name="nabla" role="OPERATOR">∇</XMTok>
                      <XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
                    </XMApp>
                    <XMTok name="Psi" role="UNKNOWN" xml:id="S1.Ex12.m1.1">Ψ</XMTok>
                  </XMApp>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex12.m1.8">
                    <XMApp>
                      <XMTok meaning="laplacian"/>
                      <XMRef idref="S1.Ex12.m1.2"/>
                    </XMApp>
                    <XMApp>
                      <XMApp xml:id="S1.Ex12.m1.4">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                        <XMTok name="nabla" role="OPERATOR">∇</XMTok>
                        <XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
                      </XMApp>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex12.m1.2">
                          <XMTok meaning="plus" role="ADDOP">+</XMTok>
                          <XMTok name="Psi" role="UNKNOWN">Ψ</XMTok>
                          <XMApp>
                            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                            <XMTok font="italic" role="UNKNOWN">X</XMTok>
                            <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                          </XMApp>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex12.m1.9">
                    <XMApp>
                      <XMTok meaning="laplacian"/>
                      <XMRef idref="S1.Ex12.m1.3"/>
                    </XMApp>
                    <XMApp>
                      <XMApp xml:id="S1.Ex12.m1.5">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                        <XMTok name="nabla" role="OPERATOR">∇</XMTok>
                        <XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
                      </XMApp>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">[</XMTok>
                        <XMApp xml:id="S1.Ex12.m1.3">
                          <XMTok meaning="plus" role="ADDOP">+</XMTok>
                          <XMTok name="Psi" role="UNKNOWN">Ψ</XMTok>
                          <XMApp>
                            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                            <XMTok font="italic" role="UNKNOWN">X</XMTok>
                            <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                          </XMApp>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">]</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
      </para>
    </subsection>
    <subsection inlist="toc" xml:id="S1.SS3">
      <tags>
        <tag>1.3</tag>
        <tag role="refnum">1.3</tag>
        <tag role="typerefnum">§1.3</tag>
      </tags>
      <title><tag close=" ">1.3</tag>Operators</title>
      <para xml:id="S1.SS3.p1">
        <equation xml:id="S1.Ex13">
          <Math mode="display" tex="\sin(\frac{X}{Y})\qquad\sin[2](x)\qquad\sin x" text="list@(sine@(X / Y), (power@(sine, 2))@(x), sine@(x))" xml:id="S1.Ex13.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex13.m1.6"/>
                  <XMRef idref="S1.Ex13.m1.7"/>
                  <XMRef idref="S1.Ex13.m1.8"/>
                </XMApp>
                <XMWrap>
                  <XMDual xml:id="S1.Ex13.m1.6">
                    <XMApp>
                      <XMTok meaning="sine"/>
                      <XMRef idref="S1.Ex13.m1.1"/>
                    </XMApp>
                    <XMApp>
                      <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.Ex13.m1.4">sin</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex13.m1.1">
                          <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex13.m1.7">
                    <XMApp>
                      <XMApp>
                        <XMTok meaning="power"/>
                        <XMTok meaning="sine"/>
                        <XMRef idref="S1.Ex13.m1.2"/>
                      </XMApp>
                      <XMRef idref="S1.Ex13.m1.3"/>
                    </XMApp>
                    <XMApp>
                      <XMApp role="OPFUNCTION" xml:id="S1.Ex13.m1.5">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                        <XMTok role="OPFUNCTION" scriptpos="post">sin</XMTok>
                        <XMTok fontsize="70%" meaning="2" role="NUMBER" xml:id="S1.Ex13.m1.2">2</XMTok>
                      </XMApp>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex13.m1.3">x</XMTok>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMApp xml:id="S1.Ex13.m1.8">
                    <XMTok meaning="sine" role="OPFUNCTION" scriptpos="post">sin</XMTok>
                    <XMTok font="italic" role="UNKNOWN">x</XMTok>
                  </XMApp>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <p>But</p>
        <equation xml:id="S1.Ex14">
          <Math mode="display" tex="\sin[\frac{X}{Y}]\qquad\sin[x][\frac{X}{Y}]\qquad\sin[x]{\frac{X}{Y}}\qquad%&#10;\sin\{\frac{X}{Y}\}\qquad\sin[x]\{\frac{X}{Y}\}" text="list@(sine@(X / Y), sine@(x) * delimited-[]@(X / Y), (power@(sine, x))@(X / Y), sine@(X / Y), sine@(x) * set@(X / Y))" xml:id="S1.Ex14.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex14.m1.15"/>
                  <XMRef idref="S1.Ex14.m1.16"/>
                  <XMRef idref="S1.Ex14.m1.14"/>
                  <XMRef idref="S1.Ex14.m1.17"/>
                  <XMRef idref="S1.Ex14.m1.18"/>
                </XMApp>
                <XMWrap>
                  <XMDual xml:id="S1.Ex14.m1.15">
                    <XMApp>
                      <XMRef idref="S1.Ex14.m1.4"/>
                      <XMRef idref="S1.Ex14.m1.5"/>
                    </XMApp>
                    <XMApp>
                      <XMTok meaning="sine" role="OPFUNCTION" scriptpos="post" xml:id="S1.Ex14.m1.4">sin</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="false">[</XMTok>
                        <XMApp xml:id="S1.Ex14.m1.5">
                          <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="false">]</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMApp xml:id="S1.Ex14.m1.16">
                    <XMTok meaning="times" role="MULOP">⁢</XMTok>
                    <XMDual>
                      <XMApp>
                        <XMRef idref="S1.Ex14.m1.6"/>
                        <XMRef idref="S1.Ex14.m1.7"/>
                      </XMApp>
                      <XMApp>
                        <XMTok meaning="sine" role="OPFUNCTION" scriptpos="post" xml:id="S1.Ex14.m1.6">sin</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="false">[</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex14.m1.7">x</XMTok>
                          <XMTok role="CLOSE" stretchy="false">]</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="delimited-[]"/>
                        <XMRef idref="S1.Ex14.m1.8"/>
                      </XMApp>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="false">[</XMTok>
                        <XMApp xml:id="S1.Ex14.m1.8">
                          <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="false">]</XMTok>
                      </XMWrap>
                    </XMDual>
                  </XMApp>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex14.m1.14">
                    <XMApp>
                      <XMApp>
                        <XMTok meaning="power"/>
                        <XMTok meaning="sine"/>
                        <XMRef idref="S1.Ex14.m1.1"/>
                      </XMApp>
                      <XMRef idref="S1.Ex14.m1.2"/>
                    </XMApp>
                    <XMApp>
                      <XMApp role="OPFUNCTION" xml:id="S1.Ex14.m1.3">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                        <XMTok role="OPFUNCTION" scriptpos="post">sin</XMTok>
                        <XMTok font="italic" fontsize="70%" role="UNKNOWN" xml:id="S1.Ex14.m1.1">x</XMTok>
                      </XMApp>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex14.m1.2">
                          <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex14.m1.17">
                    <XMApp>
                      <XMRef idref="S1.Ex14.m1.9"/>
                      <XMRef idref="S1.Ex14.m1.10"/>
                    </XMApp>
                    <XMApp>
                      <XMTok meaning="sine" role="OPFUNCTION" scriptpos="post" xml:id="S1.Ex14.m1.9">sin</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="false">{</XMTok>
                        <XMApp xml:id="S1.Ex14.m1.10">
                          <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="false">}</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMApp xml:id="S1.Ex14.m1.18">
                    <XMTok meaning="times" role="MULOP">⁢</XMTok>
                    <XMDual>
                      <XMApp>
                        <XMRef idref="S1.Ex14.m1.11"/>
                        <XMRef idref="S1.Ex14.m1.12"/>
                      </XMApp>
                      <XMApp>
                        <XMTok meaning="sine" role="OPFUNCTION" scriptpos="post" xml:id="S1.Ex14.m1.11">sin</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="false">[</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex14.m1.12">x</XMTok>
                          <XMTok role="CLOSE" stretchy="false">]</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="set"/>
                        <XMRef idref="S1.Ex14.m1.13"/>
                      </XMApp>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="false">{</XMTok>
                        <XMApp xml:id="S1.Ex14.m1.13">
                          <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="false">}</XMTok>
                      </XMWrap>
                    </XMDual>
                  </XMApp>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
      </para>
      <para xml:id="S1.SS3.p2">
        <tabular vattach="middle">
          <tbody>
            <tr>
              <td align="center"><Math mode="inline" tex="\displaystyle\sin(x)" text="sine@(x)" xml:id="S1.SS3.p2.m1">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="sine"/>
                        <XMRef idref="S1.SS3.p2.m1.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m1.2">sin</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m1.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
              <td align="center"><Math mode="inline" tex="\displaystyle\sinh(x)" text="hyperbolic-sine@(x)" xml:id="S1.SS3.p2.m2">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMRef idref="S1.SS3.p2.m2.1"/>
                        <XMRef idref="S1.SS3.p2.m2.2"/>
                      </XMApp>
                      <XMApp>
                        <XMTok meaning="hyperbolic-sine" role="TRIGFUNCTION" xml:id="S1.SS3.p2.m2.1">sinh</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="false">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m2.2">x</XMTok>
                          <XMTok role="CLOSE" stretchy="false">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
              <td align="center"><Math mode="inline" tex="\displaystyle\arcsin(x)" text="arcsine@(x)" xml:id="S1.SS3.p2.m3">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="arcsine"/>
                        <XMRef idref="S1.SS3.p2.m3.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m3.2">arcsin</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m3.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
              <td align="center"><Math mode="inline" tex="\displaystyle\asin(x)" text="arcsine@(x)" xml:id="S1.SS3.p2.m4">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="arcsine"/>
                        <XMRef idref="S1.SS3.p2.m4.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m4.2">asin</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m4.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
            </tr>
            <tr>
              <td align="center"><Math mode="inline" tex="\displaystyle\cos(x)" text="cosine@(x)" xml:id="S1.SS3.p2.m5">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="cosine"/>
                        <XMRef idref="S1.SS3.p2.m5.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m5.2">cos</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m5.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
              <td align="center"><Math mode="inline" tex="\displaystyle\cosh(x)" text="hyperbolic-cosine@(x)" xml:id="S1.SS3.p2.m6">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="hyperbolic-cosine"/>
                        <XMRef idref="S1.SS3.p2.m6.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m6.2">cosh</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m6.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
              <td align="center"><Math mode="inline" tex="\displaystyle\arccos(x)" text="arccosine@(x)" xml:id="S1.SS3.p2.m7">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="arccosine"/>
                        <XMRef idref="S1.SS3.p2.m7.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m7.2">arccos</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m7.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
              <td align="center"><Math mode="inline" tex="\displaystyle\acos(x)" text="arccosine@(x)" xml:id="S1.SS3.p2.m8">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="arccosine"/>
                        <XMRef idref="S1.SS3.p2.m8.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m8.2">acos</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m8.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
            </tr>
            <tr>
              <td align="center"><Math mode="inline" tex="\displaystyle\tan(x)" text="tangent@(x)" xml:id="S1.SS3.p2.m9">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="tangent"/>
                        <XMRef idref="S1.SS3.p2.m9.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m9.2">tan</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m9.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
              <td align="center"><Math mode="inline" tex="\displaystyle\tanh(x)" text="hyperbolic-tangent@(x)" xml:id="S1.SS3.p2.m10">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="hyperbolic-tangent"/>
                        <XMRef idref="S1.SS3.p2.m10.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m10.2">tanh</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m10.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
              <td align="center"><Math mode="inline" tex="\displaystyle\arctan(x)" text="arctangent@(x)" xml:id="S1.SS3.p2.m11">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="arctangent"/>
                        <XMRef idref="S1.SS3.p2.m11.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m11.2">arctan</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m11.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
              <td align="center"><Math mode="inline" tex="\displaystyle\atan(x)" text="arctangent@(x)" xml:id="S1.SS3.p2.m12">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="arctangent"/>
                        <XMRef idref="S1.SS3.p2.m12.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m12.2">atan</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m12.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
            </tr>
            <tr>
              <td align="center"><Math mode="inline" tex="\displaystyle\csc(x)" text="cosecant@(x)" xml:id="S1.SS3.p2.m13">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="cosecant"/>
                        <XMRef idref="S1.SS3.p2.m13.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m13.2">csc</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m13.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
              <td align="center"><Math mode="inline" tex="\displaystyle\csch(x)" text="hyperbolic-cosecant@(x)" xml:id="S1.SS3.p2.m14">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="hyperbolic-cosecant"/>
                        <XMRef idref="S1.SS3.p2.m14.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m14.2">csch</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m14.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
              <td align="center"><Math mode="inline" tex="\displaystyle\arccsc(x)" text="arccosecant@(x)" xml:id="S1.SS3.p2.m15">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="arccosecant"/>
                        <XMRef idref="S1.SS3.p2.m15.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m15.2">arccsc</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m15.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
              <td align="center"><Math mode="inline" tex="\displaystyle\acsc(x)" text="arccosecant@(x)" xml:id="S1.SS3.p2.m16">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="arccosecant"/>
                        <XMRef idref="S1.SS3.p2.m16.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m16.2">acsc</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m16.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
            </tr>
            <tr>
              <td align="center"><Math mode="inline" tex="\displaystyle\sec(x)" text="secant@(x)" xml:id="S1.SS3.p2.m17">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="secant"/>
                        <XMRef idref="S1.SS3.p2.m17.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m17.2">sec</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m17.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
              <td align="center"><Math mode="inline" tex="\displaystyle\sech(x)" text="hyperbolic-secant@(x)" xml:id="S1.SS3.p2.m18">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="hyperbolic-secant"/>
                        <XMRef idref="S1.SS3.p2.m18.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m18.2">sech</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m18.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
              <td align="center"><Math mode="inline" tex="\displaystyle\arcsec(x)" text="arcsecant@(x)" xml:id="S1.SS3.p2.m19">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="arcsecant"/>
                        <XMRef idref="S1.SS3.p2.m19.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m19.2">arcsec</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m19.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
              <td align="center"><Math mode="inline" tex="\displaystyle\asec(x)" text="arcsecant@(x)" xml:id="S1.SS3.p2.m20">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="arcsecant"/>
                        <XMRef idref="S1.SS3.p2.m20.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m20.2">asec</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m20.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
            </tr>
            <tr>
              <td align="center"><Math mode="inline" tex="\displaystyle\cot(x)" text="cotangent@(x)" xml:id="S1.SS3.p2.m21">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="cotangent"/>
                        <XMRef idref="S1.SS3.p2.m21.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m21.2">cot</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m21.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
              <td align="center"><Math mode="inline" tex="\displaystyle\coth(x)" text="hyperbolic-cotangent@(x)" xml:id="S1.SS3.p2.m22">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="hyperbolic-cotangent"/>
                        <XMRef idref="S1.SS3.p2.m22.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m22.2">coth</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m22.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
              <td align="center"><Math mode="inline" tex="\displaystyle\arccot(x)" text="arccotangent@(x)" xml:id="S1.SS3.p2.m23">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="arccotangent"/>
                        <XMRef idref="S1.SS3.p2.m23.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m23.2">arccot</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m23.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
              <td align="center"><Math mode="inline" tex="\displaystyle\acot(x)" text="arccotangent@(x)" xml:id="S1.SS3.p2.m24">
                  <XMath>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="arccotangent"/>
                        <XMRef idref="S1.SS3.p2.m24.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.SS3.p2.m24.2">acot</XMTok>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="true">(</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.SS3.p2.m24.1">x</XMTok>
                          <XMTok role="CLOSE" stretchy="true">)</XMTok>
                        </XMWrap>
                      </XMApp>
                    </XMDual>
                  </XMath>
                </Math></td>
            </tr>
          </tbody>
        </tabular>
      </para>
      <para xml:id="S1.SS3.p3">
        <equation xml:id="S1.Ex15">
          <Math mode="display" tex="\exp(X^{Y})\qquad\log(X^{Y})\qquad\ln(X^{Y})\qquad\det(X^{Y})\qquad\Pr(X^{Y})" text="list@(exponential@(X ^ Y), logarithm@(X ^ Y), natural-logarithm@(X ^ Y), determinant@(X ^ Y), probability@(X ^ Y))" xml:id="S1.Ex15.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex15.m1.11"/>
                  <XMRef idref="S1.Ex15.m1.12"/>
                  <XMRef idref="S1.Ex15.m1.13"/>
                  <XMRef idref="S1.Ex15.m1.14"/>
                  <XMRef idref="S1.Ex15.m1.15"/>
                </XMApp>
                <XMWrap>
                  <XMDual xml:id="S1.Ex15.m1.11">
                    <XMApp>
                      <XMTok meaning="exponential"/>
                      <XMRef idref="S1.Ex15.m1.1"/>
                    </XMApp>
                    <XMApp>
                      <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.Ex15.m1.6">exp</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex15.m1.1">
                          <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex15.m1.12">
                    <XMApp>
                      <XMTok meaning="logarithm"/>
                      <XMRef idref="S1.Ex15.m1.2"/>
                    </XMApp>
                    <XMApp>
                      <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.Ex15.m1.7">log</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex15.m1.2">
                          <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex15.m1.13">
                    <XMApp>
                      <XMTok meaning="natural-logarithm"/>
                      <XMRef idref="S1.Ex15.m1.3"/>
                    </XMApp>
                    <XMApp>
                      <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.Ex15.m1.8">ln</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex15.m1.3">
                          <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex15.m1.14">
                    <XMApp>
                      <XMTok meaning="determinant"/>
                      <XMRef idref="S1.Ex15.m1.4"/>
                    </XMApp>
                    <XMApp>
                      <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.Ex15.m1.9">det</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex15.m1.4">
                          <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex15.m1.15">
                    <XMApp>
                      <XMTok meaning="probability"/>
                      <XMRef idref="S1.Ex15.m1.5"/>
                    </XMApp>
                    <XMApp>
                      <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.Ex15.m1.10">Pr</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex15.m1.5">
                          <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
      </para>
      <para xml:id="S1.SS3.p4">
        <equation xml:id="S1.Ex16">
          <Math mode="display" tex="\tr\rho\qquad\tr(X^{Y})\qquad\Tr\rho\qquad\rank M\qquad\erf(x)\qquad\Res[f(z)]" text="list@(trace@(rho), trace@(X ^ Y), trace@(rho), rank@(M), error-function@(x), residue@(f * z))" xml:id="S1.Ex16.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex16.m1.9"/>
                  <XMRef idref="S1.Ex16.m1.7"/>
                  <XMRef idref="S1.Ex16.m1.10"/>
                  <XMRef idref="S1.Ex16.m1.11"/>
                  <XMRef idref="S1.Ex16.m1.8"/>
                  <XMRef idref="S1.Ex16.m1.12"/>
                </XMApp>
                <XMWrap>
                  <XMApp xml:id="S1.Ex16.m1.9">
                    <XMTok meaning="trace" role="OPFUNCTION" scriptpos="post">tr</XMTok>
                    <XMTok font="italic" name="rho" role="UNKNOWN">ρ</XMTok>
                  </XMApp>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex16.m1.7">
                    <XMApp>
                      <XMTok meaning="trace"/>
                      <XMRef idref="S1.Ex16.m1.1"/>
                    </XMApp>
                    <XMApp>
                      <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.Ex16.m1.3">tr</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex16.m1.1">
                          <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMApp xml:id="S1.Ex16.m1.10">
                    <XMTok meaning="trace" role="OPFUNCTION" scriptpos="post">Tr</XMTok>
                    <XMTok font="italic" name="rho" role="UNKNOWN">ρ</XMTok>
                  </XMApp>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMApp xml:id="S1.Ex16.m1.11">
                    <XMTok meaning="rank" role="OPFUNCTION" scriptpos="post">rank</XMTok>
                    <XMTok font="italic" role="UNKNOWN">M</XMTok>
                  </XMApp>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex16.m1.8">
                    <XMApp>
                      <XMTok meaning="error-function"/>
                      <XMRef idref="S1.Ex16.m1.2"/>
                    </XMApp>
                    <XMApp>
                      <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.Ex16.m1.4">erf</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex16.m1.2">x</XMTok>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex16.m1.12">
                    <XMApp>
                      <XMRef idref="S1.Ex16.m1.6"/>
                      <XMRef idref="S1.Ex16.m1.12.1"/>
                    </XMApp>
                    <XMApp>
                      <XMTok meaning="residue" role="OPFUNCTION" scriptpos="post" xml:id="S1.Ex16.m1.6">Res</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="false">[</XMTok>
                        <XMApp xml:id="S1.Ex16.m1.12.1">
                          <XMTok meaning="times" role="MULOP">⁢</XMTok>
                          <XMTok font="italic" role="UNKNOWN">f</XMTok>
                          <XMDual>
                            <XMRef idref="S1.Ex16.m1.5"/>
                            <XMWrap>
                              <XMTok role="OPEN" stretchy="false">(</XMTok>
                              <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex16.m1.5">z</XMTok>
                              <XMTok role="CLOSE" stretchy="false">)</XMTok>
                            </XMWrap>
                          </XMDual>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="false">]</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex17">
          <Math mode="display" tex="\qquad\principalvalue{\int f(z)\differential{z}}\qquad\PV{\int f(z)%&#10;\differential{z}}\qquad\Re{z}\qquad\Re\qquad\Im{z}\qquad\Im" text="list@(principal-value@(integral@(f * z * differential@(z))), principal-value@(integral@(f * z * differential@(z))), real-part@(z), real-part, imaginary-part@(z), imaginary-part)" xml:id="S1.Ex17.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex17.m1.10"/>
                  <XMRef idref="S1.Ex17.m1.11"/>
                  <XMRef idref="S1.Ex17.m1.12"/>
                  <XMRef idref="S1.Ex17.m1.13"/>
                  <XMRef idref="S1.Ex17.m1.14"/>
                  <XMRef idref="S1.Ex17.m1.15"/>
                </XMApp>
                <XMWrap>
                  <XMDual lpadding="20.0pt" xml:id="S1.Ex17.m1.10">
                    <XMApp>
                      <XMTok meaning="principal-value"/>
                      <XMRef idref="S1.Ex17.m1.1"/>
                    </XMApp>
                    <XMApp>
                      <XMTok meaning="times" role="MULOP">⁢</XMTok>
                      <XMTok font="caligraphic" role="UNKNOWN">P</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex17.m1.1">
                          <XMTok mathstyle="display" meaning="integral" name="int" role="INTOP">∫</XMTok>
                          <XMApp>
                            <XMTok meaning="times" role="MULOP">⁢</XMTok>
                            <XMTok font="italic" role="UNKNOWN">f</XMTok>
                            <XMDual>
                              <XMRef idref="S1.Ex17.m1.1.2"/>
                              <XMWrap>
                                <XMTok role="OPEN" stretchy="false">(</XMTok>
                                <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex17.m1.1.2">z</XMTok>
                                <XMTok role="CLOSE" stretchy="false">)</XMTok>
                              </XMWrap>
                            </XMDual>
                            <XMApp>
                              <XMTok meaning="differential" role="DIFFOP">d</XMTok>
                              <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex17.m1.1.1">z</XMTok>
                            </XMApp>
                          </XMApp>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex17.m1.11">
                    <XMApp>
                      <XMTok meaning="principal-value"/>
                      <XMRef idref="S1.Ex17.m1.2"/>
                    </XMApp>
                    <XMApp>
                      <XMWrap role="OPFUNCTION" scriptpos="post" xml:id="S1.Ex17.m1.7">
                        <XMTok role="UNKNOWN" xml:id="S1.Ex17.m1.5">P</XMTok>
                        <XMTok role="PERIOD">.</XMTok>
                        <XMTok role="UNKNOWN" xml:id="S1.Ex17.m1.6">V</XMTok>
                        <XMTok role="PERIOD">.</XMTok>
                      </XMWrap>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex17.m1.2">
                          <XMTok mathstyle="display" meaning="integral" name="int" role="INTOP">∫</XMTok>
                          <XMApp>
                            <XMTok meaning="times" role="MULOP">⁢</XMTok>
                            <XMTok font="italic" role="UNKNOWN">f</XMTok>
                            <XMDual>
                              <XMRef idref="S1.Ex17.m1.2.2"/>
                              <XMWrap>
                                <XMTok role="OPEN" stretchy="false">(</XMTok>
                                <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex17.m1.2.2">z</XMTok>
                                <XMTok role="CLOSE" stretchy="false">)</XMTok>
                              </XMWrap>
                            </XMDual>
                            <XMApp>
                              <XMTok meaning="differential" role="DIFFOP">d</XMTok>
                              <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex17.m1.2.1">z</XMTok>
                            </XMApp>
                          </XMApp>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex17.m1.12">
                    <XMApp>
                      <XMTok meaning="real-part"/>
                      <XMRef idref="S1.Ex17.m1.3"/>
                    </XMApp>
                    <XMApp>
                      <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.Ex17.m1.8">Re</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">{</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex17.m1.3">z</XMTok>
                        <XMTok role="CLOSE" stretchy="true">}</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMTok meaning="real-part" name="Re" role="OPFUNCTION" xml:id="S1.Ex17.m1.13">ℜ</XMTok>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex17.m1.14">
                    <XMApp>
                      <XMTok meaning="imaginary-part"/>
                      <XMRef idref="S1.Ex17.m1.4"/>
                    </XMApp>
                    <XMApp>
                      <XMTok role="OPFUNCTION" scriptpos="post" xml:id="S1.Ex17.m1.9">Im</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">{</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex17.m1.4">z</XMTok>
                        <XMTok role="CLOSE" stretchy="true">}</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMTok meaning="imaginary-part" name="Im" role="OPFUNCTION" xml:id="S1.Ex17.m1.15">ℑ</XMTok>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
      </para>
      <para xml:id="S1.SS3.p5">
        <p>But</p>
        <equation xml:id="S1.Ex18">
          <Math mode="display" tex="\real(\frac{X}{Y})\qquad\real[\frac{X}{Y}]\qquad\imaginary(\frac{X}{Y})\qquad%&#10;\imaginary[\frac{X}{Y}]" text="list@(real-part@(X / Y), real-part@(X / Y), imaginary-part@(X / Y), imaginary-part@(X / Y))" xml:id="S1.Ex18.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex18.m1.9"/>
                  <XMRef idref="S1.Ex18.m1.10"/>
                  <XMRef idref="S1.Ex18.m1.11"/>
                  <XMRef idref="S1.Ex18.m1.12"/>
                </XMApp>
                <XMWrap>
                  <XMDual xml:id="S1.Ex18.m1.9">
                    <XMApp>
                      <XMRef idref="S1.Ex18.m1.1"/>
                      <XMRef idref="S1.Ex18.m1.2"/>
                    </XMApp>
                    <XMApp>
                      <XMTok meaning="real-part" role="OPERATOR" scriptpos="post" xml:id="S1.Ex18.m1.1">Re</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="false">(</XMTok>
                        <XMApp xml:id="S1.Ex18.m1.2">
                          <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="false">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex18.m1.10">
                    <XMApp>
                      <XMRef idref="S1.Ex18.m1.3"/>
                      <XMRef idref="S1.Ex18.m1.4"/>
                    </XMApp>
                    <XMApp>
                      <XMTok meaning="real-part" role="OPERATOR" scriptpos="post" xml:id="S1.Ex18.m1.3">Re</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="false">[</XMTok>
                        <XMApp xml:id="S1.Ex18.m1.4">
                          <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="false">]</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex18.m1.11">
                    <XMApp>
                      <XMRef idref="S1.Ex18.m1.5"/>
                      <XMRef idref="S1.Ex18.m1.6"/>
                    </XMApp>
                    <XMApp>
                      <XMTok meaning="imaginary-part" role="OPERATOR" scriptpos="post" xml:id="S1.Ex18.m1.5">Im</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="false">(</XMTok>
                        <XMApp xml:id="S1.Ex18.m1.6">
                          <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="false">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex18.m1.12">
                    <XMApp>
                      <XMRef idref="S1.Ex18.m1.7"/>
                      <XMRef idref="S1.Ex18.m1.8"/>
                    </XMApp>
                    <XMApp>
                      <XMTok meaning="imaginary-part" role="OPERATOR" scriptpos="post" xml:id="S1.Ex18.m1.7">Im</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="false">[</XMTok>
                        <XMApp xml:id="S1.Ex18.m1.8">
                          <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="false">]</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
      </para>
    </subsection>
    <subsection inlist="toc" xml:id="S1.SS4">
      <tags>
        <tag>1.4</tag>
        <tag role="refnum">1.4</tag>
        <tag role="typerefnum">§1.4</tag>
      </tags>
      <title><tag close=" ">1.4</tag>Quick quad text</title>
      <para xml:id="S1.SS4.p1">
        <equation xml:id="S1.Ex19">
          <Math mode="display" tex="[\mbox{\quad word or phrase\quad}][\mbox{word or phrase\quad}]" text="delimited-[]@([ word or phrase ]) * delimited-[]@([word or phrase ])" xml:id="S1.Ex19.m1">
            <XMath>
              <XMApp>
                <XMTok meaning="times" role="MULOP">⁢</XMTok>
                <XMDual>
                  <XMApp>
                    <XMTok meaning="delimited-[]"/>
                    <XMRef idref="S1.Ex19.m1.1"/>
                  </XMApp>
                  <XMWrap>
                    <XMTok role="OPEN" stretchy="false">[</XMTok>
                    <XMText xml:id="S1.Ex19.m1.1"> word or phrase </XMText>
                    <XMTok role="CLOSE" stretchy="false">]</XMTok>
                  </XMWrap>
                </XMDual>
                <XMDual>
                  <XMApp>
                    <XMTok meaning="delimited-[]"/>
                    <XMRef idref="S1.Ex19.m1.2"/>
                  </XMApp>
                  <XMWrap>
                    <XMTok role="OPEN" stretchy="false">[</XMTok>
                    <XMText xml:id="S1.Ex19.m1.2">word or phrase </XMText>
                    <XMTok role="CLOSE" stretchy="false">]</XMTok>
                  </XMWrap>
                </XMDual>
              </XMApp>
            </XMath>
          </Math>
        </equation>
      </para>
      <para xml:id="S1.SS4.p2">
        <equation xml:id="S1.Ex20">
          <Math mode="display" tex="[,\quad],[\mbox{\quad c.c.\quad}],[\mbox{\quad if\quad}],[\mbox{\quad then%&#10;\quad}],[\mbox{\quad else\quad}],[\mbox{\quad otherwise\quad}],[\mbox{\quad&#10;unless%&#10;\quad}],[\mbox{\quad given\quad}]" xml:id="S1.Ex20.m1">
            <XMath>
              <XMWrap>
                <XMTok role="OPEN" stretchy="false">[</XMTok>
                <XMTok role="PUNCT" rpadding="10.0pt">,</XMTok>
                <XMTok role="CLOSE" stretchy="false">]</XMTok>
              </XMWrap>
              <XMTok role="PUNCT">,</XMTok>
              <XMWrap>
                <XMTok role="OPEN" stretchy="false">[</XMTok>
                <XMText> c.c. </XMText>
                <XMTok role="CLOSE" stretchy="false">]</XMTok>
              </XMWrap>
              <XMTok role="PUNCT">,</XMTok>
              <XMWrap>
                <XMTok role="OPEN" stretchy="false">[</XMTok>
                <XMText> if </XMText>
                <XMTok role="CLOSE" stretchy="false">]</XMTok>
              </XMWrap>
              <XMTok role="PUNCT">,</XMTok>
              <XMWrap>
                <XMTok role="OPEN" stretchy="false">[</XMTok>
                <XMText> then </XMText>
                <XMTok role="CLOSE" stretchy="false">]</XMTok>
              </XMWrap>
              <XMTok role="PUNCT">,</XMTok>
              <XMWrap>
                <XMTok role="OPEN" stretchy="false">[</XMTok>
                <XMText> else </XMText>
                <XMTok role="CLOSE" stretchy="false">]</XMTok>
              </XMWrap>
              <XMTok role="PUNCT">,</XMTok>
              <XMWrap>
                <XMTok role="OPEN" stretchy="false">[</XMTok>
                <XMText> otherwise </XMText>
                <XMTok role="CLOSE" stretchy="false">]</XMTok>
              </XMWrap>
              <XMTok role="PUNCT">,</XMTok>
              <XMWrap>
                <XMTok role="OPEN" stretchy="false">[</XMTok>
                <XMText> unless </XMText>
                <XMTok role="CLOSE" stretchy="false">]</XMTok>
              </XMWrap>
              <XMTok role="PUNCT">,</XMTok>
              <XMWrap>
                <XMTok role="OPEN" stretchy="false">[</XMTok>
                <XMText> given </XMText>
                <XMTok role="CLOSE" stretchy="false">]</XMTok>
              </XMWrap>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex21">
          <Math mode="display" tex="[\mbox{\quad using\quad}],[\mbox{\quad assume\quad}],[\mbox{\quad since\quad}]%&#10;,[\mbox{\quad let\quad}],[\mbox{\quad for\quad}],[\mbox{\quad all\quad}],[%&#10;\mbox{\quad even\quad}],[\mbox{\quad odd\quad}]," text="list@(delimited-[]@([ using ]), delimited-[]@([ assume ]), delimited-[]@([ since ]), delimited-[]@([ let ]), delimited-[]@([ for ]), delimited-[]@([ all ]), delimited-[]@([ even ]), delimited-[]@([ odd ]))" xml:id="S1.Ex21.m1">
            <XMath>
              <XMDual>
                <XMRef idref="S1.Ex21.m1.9"/>
                <XMWrap>
                  <XMDual xml:id="S1.Ex21.m1.9">
                    <XMApp>
                      <XMTok meaning="list"/>
                      <XMRef idref="S1.Ex21.m1.9.1"/>
                      <XMRef idref="S1.Ex21.m1.9.2"/>
                      <XMRef idref="S1.Ex21.m1.9.3"/>
                      <XMRef idref="S1.Ex21.m1.9.4"/>
                      <XMRef idref="S1.Ex21.m1.9.5"/>
                      <XMRef idref="S1.Ex21.m1.9.6"/>
                      <XMRef idref="S1.Ex21.m1.9.7"/>
                      <XMRef idref="S1.Ex21.m1.9.8"/>
                    </XMApp>
                    <XMWrap>
                      <XMDual xml:id="S1.Ex21.m1.9.1">
                        <XMApp>
                          <XMTok meaning="delimited-[]"/>
                          <XMRef idref="S1.Ex21.m1.1"/>
                        </XMApp>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="false">[</XMTok>
                          <XMText xml:id="S1.Ex21.m1.1"> using </XMText>
                          <XMTok role="CLOSE" stretchy="false">]</XMTok>
                        </XMWrap>
                      </XMDual>
                      <XMTok role="PUNCT">,</XMTok>
                      <XMDual xml:id="S1.Ex21.m1.9.2">
                        <XMApp>
                          <XMTok meaning="delimited-[]"/>
                          <XMRef idref="S1.Ex21.m1.2"/>
                        </XMApp>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="false">[</XMTok>
                          <XMText xml:id="S1.Ex21.m1.2"> assume </XMText>
                          <XMTok role="CLOSE" stretchy="false">]</XMTok>
                        </XMWrap>
                      </XMDual>
                      <XMTok role="PUNCT">,</XMTok>
                      <XMDual xml:id="S1.Ex21.m1.9.3">
                        <XMApp>
                          <XMTok meaning="delimited-[]"/>
                          <XMRef idref="S1.Ex21.m1.3"/>
                        </XMApp>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="false">[</XMTok>
                          <XMText xml:id="S1.Ex21.m1.3"> since </XMText>
                          <XMTok role="CLOSE" stretchy="false">]</XMTok>
                        </XMWrap>
                      </XMDual>
                      <XMTok role="PUNCT">,</XMTok>
                      <XMDual xml:id="S1.Ex21.m1.9.4">
                        <XMApp>
                          <XMTok meaning="delimited-[]"/>
                          <XMRef idref="S1.Ex21.m1.4"/>
                        </XMApp>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="false">[</XMTok>
                          <XMText xml:id="S1.Ex21.m1.4"> let </XMText>
                          <XMTok role="CLOSE" stretchy="false">]</XMTok>
                        </XMWrap>
                      </XMDual>
                      <XMTok role="PUNCT">,</XMTok>
                      <XMDual xml:id="S1.Ex21.m1.9.5">
                        <XMApp>
                          <XMTok meaning="delimited-[]"/>
                          <XMRef idref="S1.Ex21.m1.5"/>
                        </XMApp>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="false">[</XMTok>
                          <XMText xml:id="S1.Ex21.m1.5"> for </XMText>
                          <XMTok role="CLOSE" stretchy="false">]</XMTok>
                        </XMWrap>
                      </XMDual>
                      <XMTok role="PUNCT">,</XMTok>
                      <XMDual xml:id="S1.Ex21.m1.9.6">
                        <XMApp>
                          <XMTok meaning="delimited-[]"/>
                          <XMRef idref="S1.Ex21.m1.6"/>
                        </XMApp>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="false">[</XMTok>
                          <XMText xml:id="S1.Ex21.m1.6"> all </XMText>
                          <XMTok role="CLOSE" stretchy="false">]</XMTok>
                        </XMWrap>
                      </XMDual>
                      <XMTok role="PUNCT">,</XMTok>
                      <XMDual xml:id="S1.Ex21.m1.9.7">
                        <XMApp>
                          <XMTok meaning="delimited-[]"/>
                          <XMRef idref="S1.Ex21.m1.7"/>
                        </XMApp>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="false">[</XMTok>
                          <XMText xml:id="S1.Ex21.m1.7"> even </XMText>
                          <XMTok role="CLOSE" stretchy="false">]</XMTok>
                        </XMWrap>
                      </XMDual>
                      <XMTok role="PUNCT">,</XMTok>
                      <XMDual xml:id="S1.Ex21.m1.9.8">
                        <XMApp>
                          <XMTok meaning="delimited-[]"/>
                          <XMRef idref="S1.Ex21.m1.8"/>
                        </XMApp>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="false">[</XMTok>
                          <XMText xml:id="S1.Ex21.m1.8"> odd </XMText>
                          <XMTok role="CLOSE" stretchy="false">]</XMTok>
                        </XMWrap>
                      </XMDual>
                    </XMWrap>
                  </XMDual>
                  <XMTok role="PUNCT">,</XMTok>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex22">
          <Math mode="display" tex="[\mbox{\quad integer\quad}],[\mbox{\quad and\quad}],[\mbox{\quad or\quad}],[%&#10;\mbox{\quad as\quad}],[\mbox{\quad in\quad}]" text="list@(delimited-[]@([ integer ]), delimited-[]@([ and ]), delimited-[]@([ or ]), delimited-[]@([ as ]), delimited-[]@([ in ]))" xml:id="S1.Ex22.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex22.m1.6"/>
                  <XMRef idref="S1.Ex22.m1.7"/>
                  <XMRef idref="S1.Ex22.m1.8"/>
                  <XMRef idref="S1.Ex22.m1.9"/>
                  <XMRef idref="S1.Ex22.m1.10"/>
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                <XMWrap>
                  <XMDual xml:id="S1.Ex22.m1.6">
                    <XMApp>
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                      <XMRef idref="S1.Ex22.m1.1"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="false">[</XMTok>
                      <XMText xml:id="S1.Ex22.m1.1"> integer </XMText>
                      <XMTok role="CLOSE" stretchy="false">]</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok role="PUNCT">,</XMTok>
                  <XMDual xml:id="S1.Ex22.m1.7">
                    <XMApp>
                      <XMTok meaning="delimited-[]"/>
                      <XMRef idref="S1.Ex22.m1.2"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="false">[</XMTok>
                      <XMText xml:id="S1.Ex22.m1.2"> and </XMText>
                      <XMTok role="CLOSE" stretchy="false">]</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok role="PUNCT">,</XMTok>
                  <XMDual xml:id="S1.Ex22.m1.8">
                    <XMApp>
                      <XMTok meaning="delimited-[]"/>
                      <XMRef idref="S1.Ex22.m1.3"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="false">[</XMTok>
                      <XMText xml:id="S1.Ex22.m1.3"> or </XMText>
                      <XMTok role="CLOSE" stretchy="false">]</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok role="PUNCT">,</XMTok>
                  <XMDual xml:id="S1.Ex22.m1.9">
                    <XMApp>
                      <XMTok meaning="delimited-[]"/>
                      <XMRef idref="S1.Ex22.m1.4"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="false">[</XMTok>
                      <XMText xml:id="S1.Ex22.m1.4"> as </XMText>
                      <XMTok role="CLOSE" stretchy="false">]</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok role="PUNCT">,</XMTok>
                  <XMDual xml:id="S1.Ex22.m1.10">
                    <XMApp>
                      <XMTok meaning="delimited-[]"/>
                      <XMRef idref="S1.Ex22.m1.5"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="false">[</XMTok>
                      <XMText xml:id="S1.Ex22.m1.5"> in </XMText>
                      <XMTok role="CLOSE" stretchy="false">]</XMTok>
                    </XMWrap>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
      </para>
    </subsection>
    <subsection inlist="toc" xml:id="S1.SS5">
      <tags>
        <tag>1.5</tag>
        <tag role="refnum">1.5</tag>
        <tag role="typerefnum">§1.5</tag>
      </tags>
      <title><tag close=" ">1.5</tag>Derivatives</title>
      <para xml:id="S1.SS5.p1">
        <equation xml:id="S1.Ex23">
          <Math mode="display" tex="\differential\qquad\differential x\qquad\differential{x}\qquad\differential[3]%&#10;{x}\qquad\differential(\cos\theta)" text="list@(differential, differential@(x), differential@(x), differential@(x, 3), differential@(cosine@(theta)))" xml:id="S1.Ex23.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex23.m1.5"/>
                  <XMRef idref="S1.Ex23.m1.9"/>
                  <XMRef idref="S1.Ex23.m1.6"/>
                  <XMRef idref="S1.Ex23.m1.7"/>
                  <XMRef idref="S1.Ex23.m1.8"/>
                </XMApp>
                <XMWrap>
                  <XMTok meaning="differential" role="DIFFOP" xml:id="S1.Ex23.m1.5">d</XMTok>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMApp xml:id="S1.Ex23.m1.9">
                    <XMTok meaning="differential" role="DIFFOP">d</XMTok>
                    <XMTok font="italic" role="UNKNOWN">x</XMTok>
                  </XMApp>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMApp xml:id="S1.Ex23.m1.6">
                    <XMTok meaning="differential" role="DIFFOP">d</XMTok>
                    <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex23.m1.1">x</XMTok>
                  </XMApp>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex23.m1.7">
                    <XMApp>
                      <XMTok meaning="differential"/>
                      <XMRef idref="S1.Ex23.m1.3"/>
                      <XMRef idref="S1.Ex23.m1.2"/>
                    </XMApp>
                    <XMApp>
                      <XMApp role="DIFFOP">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                        <XMTok role="DIFFOP">d</XMTok>
                        <XMTok fontsize="70%" meaning="3" role="NUMBER" xml:id="S1.Ex23.m1.2">3</XMTok>
                      </XMApp>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex23.m1.3">x</XMTok>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex23.m1.8">
                    <XMApp>
                      <XMTok meaning="differential"/>
                      <XMRef idref="S1.Ex23.m1.4"/>
                    </XMApp>
                    <XMApp>
                      <XMTok role="DIFFOP">d</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex23.m1.4">
                          <XMTok meaning="cosine" role="OPFUNCTION" scriptpos="post">cos</XMTok>
                          <XMTok font="italic" name="theta" role="UNKNOWN">θ</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex24">
          <Math mode="display" tex="\derivative{x}\qquad\derivative{x}f\qquad\derivative{f}{x}\qquad\derivative[n]%&#10;{f}{x}\qquad\derivative{x}(\frac{X}{Y})\qquad\derivative*{f}{x}" text="list@(derivative@(x), (derivative@(x))@(f), (derivative@(x))@(f), (derivative@(x, n))@(f), (derivative@(x))@(X / Y), (derivative@(x))@(f))" xml:id="S1.Ex24.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex24.m1.14"/>
                  <XMRef idref="S1.Ex24.m1.19"/>
                  <XMRef idref="S1.Ex24.m1.15"/>
                  <XMRef idref="S1.Ex24.m1.16"/>
                  <XMRef idref="S1.Ex24.m1.17"/>
                  <XMRef idref="S1.Ex24.m1.18"/>
                </XMApp>
                <XMWrap>
                  <XMDual role="DIFFOP" xml:id="S1.Ex24.m1.14">
                    <XMApp>
                      <XMTok meaning="derivative"/>
                      <XMRef idref="S1.Ex24.m1.1"/>
                    </XMApp>
                    <XMApp>
                      <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                      <XMTok role="DIFFOP">d</XMTok>
                      <XMApp>
                        <XMTok role="DIFFOP">d</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex24.m1.1">x</XMTok>
                      </XMApp>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMApp xml:id="S1.Ex24.m1.19">
                    <XMDual role="DIFFOP">
                      <XMApp>
                        <XMTok meaning="derivative"/>
                        <XMRef idref="S1.Ex24.m1.2"/>
                      </XMApp>
                      <XMApp>
                        <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                        <XMTok role="DIFFOP">d</XMTok>
                        <XMApp>
                          <XMTok role="DIFFOP">d</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex24.m1.2">x</XMTok>
                        </XMApp>
                      </XMApp>
                    </XMDual>
                    <XMTok font="italic" role="UNKNOWN">f</XMTok>
                  </XMApp>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex24.m1.15">
                    <XMApp>
                      <XMApp>
                        <XMTok meaning="derivative"/>
                        <XMRef idref="S1.Ex24.m1.4"/>
                      </XMApp>
                      <XMRef idref="S1.Ex24.m1.3"/>
                    </XMApp>
                    <XMApp>
                      <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                      <XMApp>
                        <XMTok role="DIFFOP">d</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex24.m1.3">f</XMTok>
                      </XMApp>
                      <XMApp>
                        <XMTok role="DIFFOP">d</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex24.m1.4">x</XMTok>
                      </XMApp>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex24.m1.16">
                    <XMApp>
                      <XMApp>
                        <XMTok meaning="derivative"/>
                        <XMRef idref="S1.Ex24.m1.7"/>
                        <XMRef idref="S1.Ex24.m1.5"/>
                      </XMApp>
                      <XMRef idref="S1.Ex24.m1.6"/>
                    </XMApp>
                    <XMApp>
                      <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                      <XMApp>
                        <XMApp role="DIFFOP">
                          <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                          <XMTok role="DIFFOP">d</XMTok>
                          <XMTok font="italic" fontsize="70%" role="UNKNOWN" xml:id="S1.Ex24.m1.5">n</XMTok>
                        </XMApp>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex24.m1.6">f</XMTok>
                      </XMApp>
                      <XMApp>
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                        <XMApp>
                          <XMTok role="DIFFOP">d</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex24.m1.7">x</XMTok>
                        </XMApp>
                        <XMTok font="italic" fontsize="70%" role="UNKNOWN" xml:id="S1.Ex24.m1.8">n</XMTok>
                      </XMApp>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex24.m1.17">
                    <XMApp>
                      <XMApp>
                        <XMTok meaning="derivative"/>
                        <XMRef idref="S1.Ex24.m1.9"/>
                      </XMApp>
                      <XMRef idref="S1.Ex24.m1.10"/>
                    </XMApp>
                    <XMApp>
                      <XMApp xml:id="S1.Ex24.m1.13">
                        <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                        <XMTok role="DIFFOP">d</XMTok>
                        <XMApp>
                          <XMTok role="DIFFOP">d</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex24.m1.9">x</XMTok>
                        </XMApp>
                      </XMApp>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex24.m1.10">
                          <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex24.m1.18">
                    <XMApp>
                      <XMApp>
                        <XMTok meaning="derivative"/>
                        <XMRef idref="S1.Ex24.m1.12"/>
                      </XMApp>
                      <XMRef idref="S1.Ex24.m1.11"/>
                    </XMApp>
                    <XMApp>
                      <XMTok meaning="divide" role="MULOP" stretchy="true">/</XMTok>
                      <XMApp>
                        <XMTok role="DIFFOP">d</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex24.m1.11">f</XMTok>
                      </XMApp>
                      <XMApp>
                        <XMTok role="DIFFOP">d</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex24.m1.12">x</XMTok>
                      </XMApp>
                    </XMApp>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex25">
          <Math mode="display" tex="\partialderivative{x}\qquad\partialderivative{x}f\qquad\partialderivative{x}%&#10;\qquad\partialderivative{f}{x}\qquad\partialderivative[n]{f}{x}\qquad%&#10;\partialderivative{x}(\frac{X}{Y})\qquad\partialderivative*{f}{x}" text="list@(partial-derivative@(x), (partial-derivative@(x))@(f), partial-derivative@(x), (partial-derivative@(x))@(f), (partial-derivative@(x, n))@(f), (partial-derivative@(x))@(X / Y), (partial-derivative@(x))@(f))" xml:id="S1.Ex25.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex25.m1.15"/>
                  <XMRef idref="S1.Ex25.m1.21"/>
                  <XMRef idref="S1.Ex25.m1.16"/>
                  <XMRef idref="S1.Ex25.m1.17"/>
                  <XMRef idref="S1.Ex25.m1.18"/>
                  <XMRef idref="S1.Ex25.m1.19"/>
                  <XMRef idref="S1.Ex25.m1.20"/>
                </XMApp>
                <XMWrap>
                  <XMDual role="DIFFOP" xml:id="S1.Ex25.m1.15">
                    <XMApp>
                      <XMTok meaning="partial-derivative"/>
                      <XMRef idref="S1.Ex25.m1.1"/>
                    </XMApp>
                    <XMApp>
                      <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                      <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                      <XMApp>
                        <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex25.m1.1">x</XMTok>
                      </XMApp>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMApp xml:id="S1.Ex25.m1.21">
                    <XMDual role="DIFFOP">
                      <XMApp>
                        <XMTok meaning="partial-derivative"/>
                        <XMRef idref="S1.Ex25.m1.2"/>
                      </XMApp>
                      <XMApp>
                        <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                        <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                        <XMApp>
                          <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex25.m1.2">x</XMTok>
                        </XMApp>
                      </XMApp>
                    </XMDual>
                    <XMTok font="italic" role="UNKNOWN">f</XMTok>
                  </XMApp>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual role="DIFFOP" xml:id="S1.Ex25.m1.16">
                    <XMApp>
                      <XMTok meaning="partial-derivative"/>
                      <XMRef idref="S1.Ex25.m1.3"/>
                    </XMApp>
                    <XMApp>
                      <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                      <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                      <XMApp>
                        <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex25.m1.3">x</XMTok>
                      </XMApp>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex25.m1.17">
                    <XMApp>
                      <XMApp>
                        <XMTok meaning="partial-derivative"/>
                        <XMRef idref="S1.Ex25.m1.5"/>
                      </XMApp>
                      <XMRef idref="S1.Ex25.m1.4"/>
                    </XMApp>
                    <XMApp>
                      <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                      <XMApp>
                        <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex25.m1.4">f</XMTok>
                      </XMApp>
                      <XMApp>
                        <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex25.m1.5">x</XMTok>
                      </XMApp>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex25.m1.18">
                    <XMApp>
                      <XMApp>
                        <XMTok meaning="partial-derivative"/>
                        <XMRef idref="S1.Ex25.m1.8"/>
                        <XMRef idref="S1.Ex25.m1.6"/>
                      </XMApp>
                      <XMRef idref="S1.Ex25.m1.7"/>
                    </XMApp>
                    <XMApp>
                      <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                      <XMApp>
                        <XMApp role="DIFFOP">
                          <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                          <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                          <XMTok font="italic" fontsize="70%" role="UNKNOWN" xml:id="S1.Ex25.m1.6">n</XMTok>
                        </XMApp>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex25.m1.7">f</XMTok>
                      </XMApp>
                      <XMApp>
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                        <XMApp>
                          <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
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                        </XMApp>
                        <XMTok font="italic" fontsize="70%" role="UNKNOWN" xml:id="S1.Ex25.m1.9">n</XMTok>
                      </XMApp>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex25.m1.19">
                    <XMApp>
                      <XMApp>
                        <XMTok meaning="partial-derivative"/>
                        <XMRef idref="S1.Ex25.m1.10"/>
                      </XMApp>
                      <XMRef idref="S1.Ex25.m1.11"/>
                    </XMApp>
                    <XMApp>
                      <XMApp xml:id="S1.Ex25.m1.14">
                        <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                        <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                        <XMApp>
                          <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex25.m1.10">x</XMTok>
                        </XMApp>
                      </XMApp>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex25.m1.11">
                          <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex25.m1.20">
                    <XMApp>
                      <XMApp>
                        <XMTok meaning="partial-derivative"/>
                        <XMRef idref="S1.Ex25.m1.13"/>
                      </XMApp>
                      <XMRef idref="S1.Ex25.m1.12"/>
                    </XMApp>
                    <XMApp>
                      <XMTok meaning="divide" role="MULOP" stretchy="true">/</XMTok>
                      <XMApp>
                        <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex25.m1.12">f</XMTok>
                      </XMApp>
                      <XMApp>
                        <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex25.m1.13">x</XMTok>
                      </XMApp>
                    </XMApp>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex26">
          <Math mode="display" tex="\variation{F[g(x)]}\qquad\variation(E-TS)\qquad\functionalderivative{g}\qquad%&#10;\functionalderivative{F}{g}\qquad\functionalderivative{V}(E-TS)\qquad%&#10;\functionalderivative*{F}{x}" text="list@(variation@(F * delimited-[]@(g * x)), variation@(E - T * S), functional-derivative@(g), (functional-derivative@(g))@(F), (functional-derivative@(V))@(E - T * S), (functional-derivative@(x))@(F))" xml:id="S1.Ex26.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex26.m1.11"/>
                  <XMRef idref="S1.Ex26.m1.12"/>
                  <XMRef idref="S1.Ex26.m1.13"/>
                  <XMRef idref="S1.Ex26.m1.14"/>
                  <XMRef idref="S1.Ex26.m1.15"/>
                  <XMRef idref="S1.Ex26.m1.16"/>
                </XMApp>
                <XMWrap>
                  <XMApp xml:id="S1.Ex26.m1.11">
                    <XMTok font="italic" meaning="variation" name="delta" role="DIFFOP">δ</XMTok>
                    <XMApp xml:id="S1.Ex26.m1.1">
                      <XMTok meaning="times" role="MULOP">⁢</XMTok>
                      <XMTok font="italic" role="UNKNOWN">F</XMTok>
                      <XMDual>
                        <XMApp>
                          <XMTok meaning="delimited-[]"/>
                          <XMRef idref="S1.Ex26.m1.1.2"/>
                        </XMApp>
                        <XMWrap>
                          <XMTok role="OPEN" stretchy="false">[</XMTok>
                          <XMApp xml:id="S1.Ex26.m1.1.2">
                            <XMTok meaning="times" role="MULOP">⁢</XMTok>
                            <XMTok font="italic" role="UNKNOWN">g</XMTok>
                            <XMDual>
                              <XMRef idref="S1.Ex26.m1.1.1"/>
                              <XMWrap>
                                <XMTok role="OPEN" stretchy="false">(</XMTok>
                                <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex26.m1.1.1">x</XMTok>
                                <XMTok role="CLOSE" stretchy="false">)</XMTok>
                              </XMWrap>
                            </XMDual>
                          </XMApp>
                          <XMTok role="CLOSE" stretchy="false">]</XMTok>
                        </XMWrap>
                      </XMDual>
                    </XMApp>
                  </XMApp>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex26.m1.12">
                    <XMApp>
                      <XMTok meaning="variation"/>
                      <XMRef idref="S1.Ex26.m1.2"/>
                    </XMApp>
                    <XMApp>
                      <XMTok font="italic" name="delta" role="DIFFOP">δ</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex26.m1.2">
                          <XMTok meaning="minus" role="ADDOP">-</XMTok>
                          <XMTok font="italic" role="UNKNOWN">E</XMTok>
                          <XMApp>
                            <XMTok meaning="times" role="MULOP">⁢</XMTok>
                            <XMTok font="italic" role="UNKNOWN">T</XMTok>
                            <XMTok font="italic" role="UNKNOWN">S</XMTok>
                          </XMApp>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual role="DIFFOP" xml:id="S1.Ex26.m1.13">
                    <XMApp>
                      <XMTok meaning="functional-derivative"/>
                      <XMRef idref="S1.Ex26.m1.3"/>
                    </XMApp>
                    <XMApp>
                      <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                      <XMTok font="italic" name="delta" role="DIFFOP">δ</XMTok>
                      <XMApp>
                        <XMTok font="italic" name="delta" role="DIFFOP">δ</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex26.m1.3">g</XMTok>
                      </XMApp>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex26.m1.14">
                    <XMApp>
                      <XMApp>
                        <XMTok meaning="functional-derivative"/>
                        <XMRef idref="S1.Ex26.m1.5"/>
                      </XMApp>
                      <XMRef idref="S1.Ex26.m1.4"/>
                    </XMApp>
                    <XMApp>
                      <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                      <XMApp>
                        <XMTok font="italic" name="delta" role="DIFFOP">δ</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex26.m1.4">F</XMTok>
                      </XMApp>
                      <XMApp>
                        <XMTok font="italic" name="delta" role="DIFFOP">δ</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex26.m1.5">g</XMTok>
                      </XMApp>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex26.m1.15">
                    <XMApp>
                      <XMApp>
                        <XMTok meaning="functional-derivative"/>
                        <XMRef idref="S1.Ex26.m1.6"/>
                      </XMApp>
                      <XMRef idref="S1.Ex26.m1.7"/>
                    </XMApp>
                    <XMApp>
                      <XMApp xml:id="S1.Ex26.m1.10">
                        <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                        <XMTok font="italic" name="delta" role="DIFFOP">δ</XMTok>
                        <XMApp>
                          <XMTok font="italic" name="delta" role="DIFFOP">δ</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex26.m1.6">V</XMTok>
                        </XMApp>
                      </XMApp>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="true">(</XMTok>
                        <XMApp xml:id="S1.Ex26.m1.7">
                          <XMTok meaning="minus" role="ADDOP">-</XMTok>
                          <XMTok font="italic" role="UNKNOWN">E</XMTok>
                          <XMApp>
                            <XMTok meaning="times" role="MULOP">⁢</XMTok>
                            <XMTok font="italic" role="UNKNOWN">T</XMTok>
                            <XMTok font="italic" role="UNKNOWN">S</XMTok>
                          </XMApp>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="true">)</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex26.m1.16">
                    <XMApp>
                      <XMApp>
                        <XMTok meaning="functional-derivative"/>
                        <XMRef idref="S1.Ex26.m1.9"/>
                      </XMApp>
                      <XMRef idref="S1.Ex26.m1.8"/>
                    </XMApp>
                    <XMApp>
                      <XMTok meaning="divide" role="MULOP" stretchy="true">/</XMTok>
                      <XMApp>
                        <XMTok font="italic" name="delta" role="DIFFOP">δ</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex26.m1.8">F</XMTok>
                      </XMApp>
                      <XMApp>
                        <XMTok font="italic" name="delta" role="DIFFOP">δ</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex26.m1.9">x</XMTok>
                      </XMApp>
                    </XMApp>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <p>But</p>
        <equation xml:id="S1.Ex27">
          <Math mode="display" tex="\differential[2][\frac{X}{Y}]" text="(functional-power@(differential, 2))@(delimited-[]@(X / Y))" xml:id="S1.Ex27.m1">
            <XMath>
              <XMApp>
                <XMApp role="DIFFOP">
                  <XMTok meaning="functional-power" role="SUPERSCRIPTOP" scriptpos="post1"/>
                  <XMTok meaning="differential" role="DIFFOP">d</XMTok>
                  <XMTok fontsize="70%" meaning="2" role="NUMBER" xml:id="S1.Ex27.m1.1">2</XMTok>
                </XMApp>
                <XMDual>
                  <XMApp>
                    <XMTok meaning="delimited-[]"/>
                    <XMRef idref="S1.Ex27.m1.2"/>
                  </XMApp>
                  <XMWrap>
                    <XMTok role="OPEN" stretchy="false">[</XMTok>
                    <XMApp xml:id="S1.Ex27.m1.2">
                      <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                      <XMTok font="italic" role="UNKNOWN">X</XMTok>
                      <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                    </XMApp>
                    <XMTok role="CLOSE" stretchy="false">]</XMTok>
                  </XMWrap>
                </XMDual>
              </XMApp>
            </XMath>
          </Math>
        </equation>
        <p>And multiple derivatives, sorta; But only for partial:</p>
        <equation xml:id="S1.Ex28">
          <Math mode="display" tex="\partialderivative{f}{x}{y}\qquad\partialderivative{f}{x}{y}{z}\qquad%&#10;\partialderivative{f}{x}{y}{z}\qquad\partialderivative{x}{y}" text="list@((partial-derivative@(x, 1, y, 1))@(f), (partial-derivative@(x, 1, y, 1))@(f) * z, (partial-derivative@(x, 1, y, 1))@(f) * z, (partial-derivative@(y))@(x))" xml:id="S1.Ex28.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex28.m1.12"/>
                  <XMRef idref="S1.Ex28.m1.14"/>
                  <XMRef idref="S1.Ex28.m1.15"/>
                  <XMRef idref="S1.Ex28.m1.13"/>
                </XMApp>
                <XMWrap>
                  <XMDual xml:id="S1.Ex28.m1.12">
                    <XMApp>
                      <XMApp>
                        <XMTok meaning="partial-derivative"/>
                        <XMRef idref="S1.Ex28.m1.2"/>
                        <XMTok meaning="1" role="NUMBER">1</XMTok>
                        <XMRef idref="S1.Ex28.m1.3"/>
                        <XMTok meaning="1" role="NUMBER">1</XMTok>
                      </XMApp>
                      <XMRef idref="S1.Ex28.m1.1"/>
                    </XMApp>
                    <XMApp>
                      <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                      <XMApp>
                        <XMApp role="DIFFOP">
                          <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                          <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                          <XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
                        </XMApp>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex28.m1.1">f</XMTok>
                      </XMApp>
                      <XMApp>
                        <XMTok meaning="times" role="MULOP">⁢</XMTok>
                        <XMApp>
                          <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex28.m1.2">x</XMTok>
                        </XMApp>
                        <XMApp>
                          <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex28.m1.3">y</XMTok>
                        </XMApp>
                      </XMApp>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMApp xml:id="S1.Ex28.m1.14">
                    <XMTok meaning="times" role="MULOP">⁢</XMTok>
                    <XMDual>
                      <XMApp>
                        <XMApp>
                          <XMTok meaning="partial-derivative"/>
                          <XMRef idref="S1.Ex28.m1.5"/>
                          <XMTok meaning="1" role="NUMBER">1</XMTok>
                          <XMRef idref="S1.Ex28.m1.6"/>
                          <XMTok meaning="1" role="NUMBER">1</XMTok>
                        </XMApp>
                        <XMRef idref="S1.Ex28.m1.4"/>
                      </XMApp>
                      <XMApp>
                        <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                        <XMApp>
                          <XMApp role="DIFFOP">
                            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                            <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                            <XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
                          </XMApp>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex28.m1.4">f</XMTok>
                        </XMApp>
                        <XMApp>
                          <XMTok meaning="times" role="MULOP">⁢</XMTok>
                          <XMApp>
                            <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                            <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex28.m1.5">x</XMTok>
                          </XMApp>
                          <XMApp>
                            <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                            <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex28.m1.6">y</XMTok>
                          </XMApp>
                        </XMApp>
                      </XMApp>
                    </XMDual>
                    <XMTok font="italic" role="UNKNOWN">z</XMTok>
                  </XMApp>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMApp xml:id="S1.Ex28.m1.15">
                    <XMTok meaning="times" role="MULOP">⁢</XMTok>
                    <XMDual>
                      <XMApp>
                        <XMApp>
                          <XMTok meaning="partial-derivative"/>
                          <XMRef idref="S1.Ex28.m1.8"/>
                          <XMTok meaning="1" role="NUMBER">1</XMTok>
                          <XMRef idref="S1.Ex28.m1.9"/>
                          <XMTok meaning="1" role="NUMBER">1</XMTok>
                        </XMApp>
                        <XMRef idref="S1.Ex28.m1.7"/>
                      </XMApp>
                      <XMApp>
                        <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                        <XMApp>
                          <XMApp role="DIFFOP">
                            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                            <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                            <XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
                          </XMApp>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex28.m1.7">f</XMTok>
                        </XMApp>
                        <XMApp>
                          <XMTok meaning="times" role="MULOP">⁢</XMTok>
                          <XMApp>
                            <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                            <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex28.m1.8">x</XMTok>
                          </XMApp>
                          <XMApp>
                            <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                            <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex28.m1.9">y</XMTok>
                          </XMApp>
                        </XMApp>
                      </XMApp>
                    </XMDual>
                    <XMTok font="italic" role="UNKNOWN">z</XMTok>
                  </XMApp>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex28.m1.13">
                    <XMApp>
                      <XMApp>
                        <XMTok meaning="partial-derivative"/>
                        <XMRef idref="S1.Ex28.m1.11"/>
                      </XMApp>
                      <XMRef idref="S1.Ex28.m1.10"/>
                    </XMApp>
                    <XMApp>
                      <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                      <XMApp>
                        <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex28.m1.10">x</XMTok>
                      </XMApp>
                      <XMApp>
                        <XMTok meaning="partial-differential" name="partial" role="DIFFOP">∂</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex28.m1.11">y</XMTok>
                      </XMApp>
                    </XMApp>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex29">
          <Math mode="display" tex="\qquad\derivative{f}{x}{y}\qquad\functionalderivative{F}{f}{g}" text="list@((derivative@(x))@(f) * y, (functional-derivative@(f))@(F) * g)" xml:id="S1.Ex29.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex29.m1.5"/>
                  <XMRef idref="S1.Ex29.m1.6"/>
                </XMApp>
                <XMWrap>
                  <XMApp xml:id="S1.Ex29.m1.5">
                    <XMTok meaning="times" role="MULOP">⁢</XMTok>
                    <XMDual lpadding="20.0pt">
                      <XMApp>
                        <XMApp>
                          <XMTok meaning="derivative"/>
                          <XMRef idref="S1.Ex29.m1.2"/>
                        </XMApp>
                        <XMRef idref="S1.Ex29.m1.1"/>
                      </XMApp>
                      <XMApp>
                        <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                        <XMApp>
                          <XMTok role="DIFFOP">d</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex29.m1.1">f</XMTok>
                        </XMApp>
                        <XMApp>
                          <XMTok role="DIFFOP">d</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex29.m1.2">x</XMTok>
                        </XMApp>
                      </XMApp>
                    </XMDual>
                    <XMTok font="italic" role="UNKNOWN">y</XMTok>
                  </XMApp>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMApp xml:id="S1.Ex29.m1.6">
                    <XMTok meaning="times" role="MULOP">⁢</XMTok>
                    <XMDual>
                      <XMApp>
                        <XMApp>
                          <XMTok meaning="functional-derivative"/>
                          <XMRef idref="S1.Ex29.m1.4"/>
                        </XMApp>
                        <XMRef idref="S1.Ex29.m1.3"/>
                      </XMApp>
                      <XMApp>
                        <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                        <XMApp>
                          <XMTok font="italic" name="delta" role="DIFFOP">δ</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex29.m1.3">F</XMTok>
                        </XMApp>
                        <XMApp>
                          <XMTok font="italic" name="delta" role="DIFFOP">δ</XMTok>
                          <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex29.m1.4">f</XMTok>
                        </XMApp>
                      </XMApp>
                    </XMDual>
                    <XMTok font="italic" role="UNKNOWN">g</XMTok>
                  </XMApp>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
      </para>
    </subsection>
    <subsection inlist="toc" xml:id="S1.SS6">
      <tags>
        <tag>1.6</tag>
        <tag role="refnum">1.6</tag>
        <tag role="typerefnum">§1.6</tag>
      </tags>
      <title><tag close=" ">1.6</tag>Dirac bra-ket notation</title>
      <para xml:id="S1.SS6.p1">
        <equation xml:id="S1.Ex30">
          <Math mode="display" tex="\bra{\phi}\ket{\psi}\mbox{\quad as opposed to\quad}\bra{\phi}\ket{\psi}" text="inner-product@(phi, psi) * [ as opposed to ] * bra@(phi) * ket@(psi)" xml:id="S1.Ex30.m1">
            <XMath>
              <XMApp>
                <XMTok meaning="times" role="MULOP">⁢</XMTok>
                <XMDual>
                  <XMApp>
                    <XMTok meaning="inner-product"/>
                    <XMRef idref="S1.Ex30.m1.1"/>
                    <XMRef idref="S1.Ex30.m1.2"/>
                  </XMApp>
                  <XMWrap>
                    <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                    <XMTok font="italic" name="phi" role="UNKNOWN" xml:id="S1.Ex30.m1.1">ϕ</XMTok>
                    <XMTok role="MIDDLE" stretchy="true">|</XMTok>
                    <XMTok font="italic" name="psi" role="UNKNOWN" xml:id="S1.Ex30.m1.2">ψ</XMTok>
                    <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                  </XMWrap>
                </XMDual>
                <XMText> as opposed to </XMText>
                <XMDual>
                  <XMApp>
                    <XMTok meaning="bra"/>
                    <XMRef idref="S1.Ex30.m1.3"/>
                  </XMApp>
                  <XMWrap>
                    <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                    <XMTok font="italic" name="phi" role="UNKNOWN" xml:id="S1.Ex30.m1.3">ϕ</XMTok>
                    <XMTok role="CLOSE" stretchy="true">|</XMTok>
                  </XMWrap>
                </XMDual>
                <XMDual>
                  <XMApp>
                    <XMTok meaning="ket"/>
                    <XMRef idref="S1.Ex30.m1.4"/>
                  </XMApp>
                  <XMWrap>
                    <XMTok role="OPEN" stretchy="true">|</XMTok>
                    <XMTok font="italic" name="psi" role="UNKNOWN" xml:id="S1.Ex30.m1.4">ψ</XMTok>
                    <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                  </XMWrap>
                </XMDual>
              </XMApp>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex31">
          <Math mode="display" tex="\bra{\phi}\outerproduct{\psi}{\xi}.\mbox{\quad as opposed to\quad}\bra{\phi}%&#10;\ket{\psi}\bra{\xi}" text="formulae@(bra@(phi) * outer-product@(psi, xi), [ as opposed to ] * inner-product@(phi, psi) * bra@(xi))" xml:id="S1.Ex31.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="formulae"/>
                  <XMRef idref="S1.Ex31.m1.7"/>
                  <XMRef idref="S1.Ex31.m1.8"/>
                </XMApp>
                <XMWrap>
                  <XMApp xml:id="S1.Ex31.m1.7">
                    <XMTok meaning="times" role="MULOP">⁢</XMTok>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="bra"/>
                        <XMRef idref="S1.Ex31.m1.1"/>
                      </XMApp>
                      <XMWrap>
                        <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                        <XMTok font="italic" name="phi" role="UNKNOWN" xml:id="S1.Ex31.m1.1">ϕ</XMTok>
                        <XMTok role="CLOSE" stretchy="true">|</XMTok>
                      </XMWrap>
                    </XMDual>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="outer-product"/>
                        <XMRef idref="S1.Ex31.m1.2"/>
                        <XMRef idref="S1.Ex31.m1.3"/>
                      </XMApp>
                      <XMWrap>
                        <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                        <XMTok font="italic" name="psi" role="UNKNOWN" xml:id="S1.Ex31.m1.2">ψ</XMTok>
                        <XMTok name="rangle" role="MIDDLE" stretchy="true">⟩</XMTok>
                        <XMWrap>
                          <XMTok name="langle" role="OPEN" stretchy="false">⟨</XMTok>
                          <XMTok font="italic" name="xi" role="UNKNOWN" xml:id="S1.Ex31.m1.3">ξ</XMTok>
                          <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                        </XMWrap>
                      </XMWrap>
                    </XMDual>
                  </XMApp>
                  <XMTok role="PERIOD">.</XMTok>
                  <XMApp xml:id="S1.Ex31.m1.8">
                    <XMTok meaning="times" role="MULOP">⁢</XMTok>
                    <XMText> as opposed to </XMText>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="inner-product"/>
                        <XMRef idref="S1.Ex31.m1.4"/>
                        <XMRef idref="S1.Ex31.m1.5"/>
                      </XMApp>
                      <XMWrap>
                        <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                        <XMTok font="italic" name="phi" role="UNKNOWN" xml:id="S1.Ex31.m1.4">ϕ</XMTok>
                        <XMTok role="MIDDLE" stretchy="true">|</XMTok>
                        <XMTok font="italic" name="psi" role="UNKNOWN" xml:id="S1.Ex31.m1.5">ψ</XMTok>
                        <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                      </XMWrap>
                    </XMDual>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="bra"/>
                        <XMRef idref="S1.Ex31.m1.6"/>
                      </XMApp>
                      <XMWrap>
                        <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                        <XMTok font="italic" name="xi" role="UNKNOWN" xml:id="S1.Ex31.m1.6">ξ</XMTok>
                        <XMTok role="CLOSE" stretchy="true">|</XMTok>
                      </XMWrap>
                    </XMDual>
                  </XMApp>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
      </para>
      <para xml:id="S1.SS6.p2">
        <equation xml:id="S1.Ex32">
          <Math mode="display" tex="\ket{X^{Y}}\qquad\ket*{X^{Y}}\qquad\bra{X^{Y}}\qquad\bra*{X^{Y}}" text="list@(ket@(X ^ Y), ket@(X ^ Y), bra@(X ^ Y), bra@(X ^ Y))" xml:id="S1.Ex32.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex32.m1.5"/>
                  <XMRef idref="S1.Ex32.m1.6"/>
                  <XMRef idref="S1.Ex32.m1.7"/>
                  <XMRef idref="S1.Ex32.m1.8"/>
                </XMApp>
                <XMWrap>
                  <XMDual xml:id="S1.Ex32.m1.5">
                    <XMApp>
                      <XMTok meaning="ket"/>
                      <XMRef idref="S1.Ex32.m1.1"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">|</XMTok>
                      <XMApp xml:id="S1.Ex32.m1.1">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex32.m1.6">
                    <XMApp>
                      <XMTok meaning="ket"/>
                      <XMRef idref="S1.Ex32.m1.2"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="false">|</XMTok>
                      <XMApp xml:id="S1.Ex32.m1.2">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok name="rangle" role="CLOSE" stretchy="false">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex32.m1.7">
                    <XMApp>
                      <XMTok meaning="bra"/>
                      <XMRef idref="S1.Ex32.m1.3"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                      <XMApp xml:id="S1.Ex32.m1.3">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok role="CLOSE" stretchy="true">|</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex32.m1.8">
                    <XMApp>
                      <XMTok meaning="bra"/>
                      <XMRef idref="S1.Ex32.m1.4"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="false">⟨</XMTok>
                      <XMApp xml:id="S1.Ex32.m1.4">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok role="CLOSE" stretchy="false">|</XMTok>
                    </XMWrap>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex33">
          <Math mode="display" tex="\bra{\phi}\ket{\psi}\qquad\bra{\phi}\ket{X^{Y}}\qquad\bra*{\phi}\ket*{X^{Y}}%&#10;\qquad\bra*{\phi}\ket*{X^{Y}}\qquad\bra*{\phi}\ket*{X^{Y}}" text="list@(inner-product@(phi, psi), inner-product@(phi, X ^ Y), inner-product@(phi, X ^ Y), inner-product@(phi, X ^ Y), inner-product@(phi, X ^ Y))" xml:id="S1.Ex33.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex33.m1.11"/>
                  <XMRef idref="S1.Ex33.m1.12"/>
                  <XMRef idref="S1.Ex33.m1.13"/>
                  <XMRef idref="S1.Ex33.m1.14"/>
                  <XMRef idref="S1.Ex33.m1.15"/>
                </XMApp>
                <XMWrap>
                  <XMDual xml:id="S1.Ex33.m1.11">
                    <XMApp>
                      <XMTok meaning="inner-product"/>
                      <XMRef idref="S1.Ex33.m1.1"/>
                      <XMRef idref="S1.Ex33.m1.2"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                      <XMTok font="italic" name="phi" role="UNKNOWN" xml:id="S1.Ex33.m1.1">ϕ</XMTok>
                      <XMTok role="MIDDLE" stretchy="true">|</XMTok>
                      <XMTok font="italic" name="psi" role="UNKNOWN" xml:id="S1.Ex33.m1.2">ψ</XMTok>
                      <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex33.m1.12">
                    <XMApp>
                      <XMTok meaning="inner-product"/>
                      <XMRef idref="S1.Ex33.m1.3"/>
                      <XMRef idref="S1.Ex33.m1.4"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                      <XMTok font="italic" name="phi" role="UNKNOWN" xml:id="S1.Ex33.m1.3">ϕ</XMTok>
                      <XMTok role="MIDDLE" stretchy="true">|</XMTok>
                      <XMApp xml:id="S1.Ex33.m1.4">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex33.m1.13">
                    <XMApp>
                      <XMTok meaning="inner-product"/>
                      <XMRef idref="S1.Ex33.m1.5"/>
                      <XMRef idref="S1.Ex33.m1.6"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="false">⟨</XMTok>
                      <XMTok font="italic" name="phi" role="UNKNOWN" xml:id="S1.Ex33.m1.5">ϕ</XMTok>
                      <XMTok role="MIDDLE" stretchy="false">|</XMTok>
                      <XMApp xml:id="S1.Ex33.m1.6">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok name="rangle" role="CLOSE" stretchy="false">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex33.m1.14">
                    <XMApp>
                      <XMTok meaning="inner-product"/>
                      <XMRef idref="S1.Ex33.m1.7"/>
                      <XMRef idref="S1.Ex33.m1.8"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="false">⟨</XMTok>
                      <XMTok font="italic" name="phi" role="UNKNOWN" xml:id="S1.Ex33.m1.7">ϕ</XMTok>
                      <XMTok role="MIDDLE" stretchy="false">|</XMTok>
                      <XMApp xml:id="S1.Ex33.m1.8">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok name="rangle" role="CLOSE" stretchy="false">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex33.m1.15">
                    <XMApp>
                      <XMTok meaning="inner-product"/>
                      <XMRef idref="S1.Ex33.m1.9"/>
                      <XMRef idref="S1.Ex33.m1.10"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="false">⟨</XMTok>
                      <XMTok font="italic" name="phi" role="UNKNOWN" xml:id="S1.Ex33.m1.9">ϕ</XMTok>
                      <XMTok role="MIDDLE" stretchy="false">|</XMTok>
                      <XMApp xml:id="S1.Ex33.m1.10">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok name="rangle" role="CLOSE" stretchy="false">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex34">
          <Math mode="display" tex="\innerproduct{a}{b}\qquad\innerproduct{a}{a}\qquad\innerproduct{a}{X^{Y}}%&#10;\qquad\innerproduct*{a}{X^{Y}}" text="list@(inner-product@(a, b), inner-product@(a, a), inner-product@(a, X ^ Y), inner-product@(a, X ^ Y))" xml:id="S1.Ex34.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex34.m1.9"/>
                  <XMRef idref="S1.Ex34.m1.10"/>
                  <XMRef idref="S1.Ex34.m1.11"/>
                  <XMRef idref="S1.Ex34.m1.12"/>
                </XMApp>
                <XMWrap>
                  <XMDual xml:id="S1.Ex34.m1.9">
                    <XMApp>
                      <XMTok meaning="inner-product"/>
                      <XMRef idref="S1.Ex34.m1.1"/>
                      <XMRef idref="S1.Ex34.m1.2"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex34.m1.1">a</XMTok>
                      <XMTok role="MIDDLE" stretchy="true">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex34.m1.2">b</XMTok>
                      <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex34.m1.10">
                    <XMApp>
                      <XMTok meaning="inner-product"/>
                      <XMRef idref="S1.Ex34.m1.3"/>
                      <XMRef idref="S1.Ex34.m1.4"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex34.m1.3">a</XMTok>
                      <XMTok role="MIDDLE" stretchy="true">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex34.m1.4">a</XMTok>
                      <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex34.m1.11">
                    <XMApp>
                      <XMTok meaning="inner-product"/>
                      <XMRef idref="S1.Ex34.m1.5"/>
                      <XMRef idref="S1.Ex34.m1.6"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex34.m1.5">a</XMTok>
                      <XMTok role="MIDDLE" stretchy="true">|</XMTok>
                      <XMApp xml:id="S1.Ex34.m1.6">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex34.m1.12">
                    <XMApp>
                      <XMTok meaning="inner-product"/>
                      <XMRef idref="S1.Ex34.m1.7"/>
                      <XMRef idref="S1.Ex34.m1.8"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="false">⟨</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex34.m1.7">a</XMTok>
                      <XMTok role="MIDDLE" stretchy="false">|</XMTok>
                      <XMApp xml:id="S1.Ex34.m1.8">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok name="rangle" role="CLOSE" stretchy="false">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex35">
          <Math mode="display" tex="\innerproduct{a}{b}\qquad\outerproduct{a}{b}\qquad\outerproduct{a}{a}\qquad%&#10;\outerproduct{a}{X^{Y}}\qquad\outerproduct*{a}{X^{Y}}\qquad\outerproduct{a}{b}" text="list@(inner-product@(a, b), outer-product@(a, b), outer-product@(a, a), outer-product@(a, X ^ Y), outer-product@(a, X ^ Y), outer-product@(a, b))" xml:id="S1.Ex35.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex35.m1.13"/>
                  <XMRef idref="S1.Ex35.m1.14"/>
                  <XMRef idref="S1.Ex35.m1.15"/>
                  <XMRef idref="S1.Ex35.m1.16"/>
                  <XMRef idref="S1.Ex35.m1.17"/>
                  <XMRef idref="S1.Ex35.m1.18"/>
                </XMApp>
                <XMWrap>
                  <XMDual xml:id="S1.Ex35.m1.13">
                    <XMApp>
                      <XMTok meaning="inner-product"/>
                      <XMRef idref="S1.Ex35.m1.1"/>
                      <XMRef idref="S1.Ex35.m1.2"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex35.m1.1">a</XMTok>
                      <XMTok role="MIDDLE" stretchy="true">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex35.m1.2">b</XMTok>
                      <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex35.m1.14">
                    <XMApp>
                      <XMTok meaning="outer-product"/>
                      <XMRef idref="S1.Ex35.m1.3"/>
                      <XMRef idref="S1.Ex35.m1.4"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex35.m1.3">a</XMTok>
                      <XMTok name="rangle" role="MIDDLE" stretchy="true">⟩</XMTok>
                      <XMWrap>
                        <XMTok name="langle" role="OPEN" stretchy="false">⟨</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex35.m1.4">b</XMTok>
                        <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                      </XMWrap>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex35.m1.15">
                    <XMApp>
                      <XMTok meaning="outer-product"/>
                      <XMRef idref="S1.Ex35.m1.5"/>
                      <XMRef idref="S1.Ex35.m1.6"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex35.m1.5">a</XMTok>
                      <XMTok name="rangle" role="MIDDLE" stretchy="true">⟩</XMTok>
                      <XMWrap>
                        <XMTok name="langle" role="OPEN" stretchy="false">⟨</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex35.m1.6">a</XMTok>
                        <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                      </XMWrap>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex35.m1.16">
                    <XMApp>
                      <XMTok meaning="outer-product"/>
                      <XMRef idref="S1.Ex35.m1.7"/>
                      <XMRef idref="S1.Ex35.m1.8"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex35.m1.7">a</XMTok>
                      <XMTok name="rangle" role="MIDDLE" stretchy="true">⟩</XMTok>
                      <XMWrap>
                        <XMTok name="langle" role="OPEN" stretchy="false">⟨</XMTok>
                        <XMApp xml:id="S1.Ex35.m1.8">
                          <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                      </XMWrap>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex35.m1.17">
                    <XMApp>
                      <XMTok meaning="outer-product"/>
                      <XMRef idref="S1.Ex35.m1.9"/>
                      <XMRef idref="S1.Ex35.m1.10"/>
                    </XMApp>
                    <XMApp>
                      <XMTok meaning="times" role="MULOP">⁢</XMTok>
                      <XMWrap>
                        <XMTok role="OPEN" stretchy="false">|</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex35.m1.9">a</XMTok>
                        <XMTok name="rangle" role="CLOSE" stretchy="false">⟩</XMTok>
                      </XMWrap>
                      <XMWrap>
                        <XMTok name="langle" role="OPEN" stretchy="false">⟨</XMTok>
                        <XMApp xml:id="S1.Ex35.m1.10">
                          <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                          <XMTok font="italic" role="UNKNOWN">X</XMTok>
                          <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                        </XMApp>
                        <XMTok role="CLOSE" stretchy="false">|</XMTok>
                      </XMWrap>
                    </XMApp>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex35.m1.18">
                    <XMApp>
                      <XMTok meaning="outer-product"/>
                      <XMRef idref="S1.Ex35.m1.11"/>
                      <XMRef idref="S1.Ex35.m1.12"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex35.m1.11">a</XMTok>
                      <XMTok name="rangle" role="MIDDLE" stretchy="true">⟩</XMTok>
                      <XMWrap>
                        <XMTok name="langle" role="OPEN" stretchy="false">⟨</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex35.m1.12">b</XMTok>
                        <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                      </XMWrap>
                    </XMWrap>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex36">
          <Math mode="display" tex="\outerproduct{a}{b}\qquad\expectationvalue{A}\qquad\expectationvalue{A}{\Psi}%&#10;\qquad\expectationvalue{A}{\Psi}\qquad\expectationvalue{\frac{X}{Y}}{\Psi}%&#10;\qquad\expectationvalue*{\frac{X}{Y}}{X^{Y}}\qquad\expectationvalue{\frac{X}{Y%&#10;}}{\Psi}" text="list@(outer-product@(a, b), expectation-value@(A), expectation-value@(A, Psi, Psi), expectation-value@(A, Psi, Psi), expectation-value@(X / Y, Psi, Psi), expectation-value@(X / Y, X ^ Y, X ^ Y), expectation-value@(X / Y, Psi, Psi))" xml:id="S1.Ex36.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex36.m1.19"/>
                  <XMRef idref="S1.Ex36.m1.20"/>
                  <XMRef idref="S1.Ex36.m1.21"/>
                  <XMRef idref="S1.Ex36.m1.22"/>
                  <XMRef idref="S1.Ex36.m1.23"/>
                  <XMRef idref="S1.Ex36.m1.24"/>
                  <XMRef idref="S1.Ex36.m1.25"/>
                </XMApp>
                <XMWrap>
                  <XMDual xml:id="S1.Ex36.m1.19">
                    <XMApp>
                      <XMTok meaning="outer-product"/>
                      <XMRef idref="S1.Ex36.m1.1"/>
                      <XMRef idref="S1.Ex36.m1.2"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex36.m1.1">a</XMTok>
                      <XMTok name="rangle" role="MIDDLE" stretchy="true">⟩</XMTok>
                      <XMWrap>
                        <XMTok name="langle" role="OPEN" stretchy="false">⟨</XMTok>
                        <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex36.m1.2">b</XMTok>
                        <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                      </XMWrap>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex36.m1.20">
                    <XMApp>
                      <XMTok meaning="expectation-value"/>
                      <XMRef idref="S1.Ex36.m1.3"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex36.m1.3">A</XMTok>
                      <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex36.m1.21">
                    <XMApp>
                      <XMTok meaning="expectation-value"/>
                      <XMRef idref="S1.Ex36.m1.5"/>
                      <XMRef idref="S1.Ex36.m1.4"/>
                      <XMRef idref="S1.Ex36.m1.6"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                      <XMTok name="Psi" role="UNKNOWN" xml:id="S1.Ex36.m1.4">Ψ</XMTok>
                      <XMTok role="CLOSE" stretchy="true">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex36.m1.5">A</XMTok>
                      <XMTok role="OPEN" stretchy="true">|</XMTok>
                      <XMTok name="Psi" role="UNKNOWN" xml:id="S1.Ex36.m1.6">Ψ</XMTok>
                      <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex36.m1.22">
                    <XMApp>
                      <XMTok meaning="expectation-value"/>
                      <XMRef idref="S1.Ex36.m1.8"/>
                      <XMRef idref="S1.Ex36.m1.7"/>
                      <XMRef idref="S1.Ex36.m1.9"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                      <XMTok name="Psi" role="UNKNOWN" xml:id="S1.Ex36.m1.7">Ψ</XMTok>
                      <XMTok role="CLOSE" stretchy="true">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex36.m1.8">A</XMTok>
                      <XMTok role="OPEN" stretchy="true">|</XMTok>
                      <XMTok name="Psi" role="UNKNOWN" xml:id="S1.Ex36.m1.9">Ψ</XMTok>
                      <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex36.m1.23">
                    <XMApp>
                      <XMTok meaning="expectation-value"/>
                      <XMRef idref="S1.Ex36.m1.11"/>
                      <XMRef idref="S1.Ex36.m1.10"/>
                      <XMRef idref="S1.Ex36.m1.12"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                      <XMTok name="Psi" role="UNKNOWN" xml:id="S1.Ex36.m1.10">Ψ</XMTok>
                      <XMTok role="CLOSE" stretchy="true">|</XMTok>
                      <XMApp xml:id="S1.Ex36.m1.11">
                        <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok role="OPEN" stretchy="true">|</XMTok>
                      <XMTok name="Psi" role="UNKNOWN" xml:id="S1.Ex36.m1.12">Ψ</XMTok>
                      <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex36.m1.24">
                    <XMApp>
                      <XMTok meaning="expectation-value"/>
                      <XMRef idref="S1.Ex36.m1.14"/>
                      <XMRef idref="S1.Ex36.m1.13"/>
                      <XMRef idref="S1.Ex36.m1.15"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="false">⟨</XMTok>
                      <XMApp xml:id="S1.Ex36.m1.13">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok role="CLOSE" stretchy="false">|</XMTok>
                      <XMApp xml:id="S1.Ex36.m1.14">
                        <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok role="OPEN" stretchy="false">|</XMTok>
                      <XMApp xml:id="S1.Ex36.m1.15">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok name="rangle" role="CLOSE" stretchy="false">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex36.m1.25">
                    <XMApp>
                      <XMTok meaning="expectation-value"/>
                      <XMRef idref="S1.Ex36.m1.17"/>
                      <XMRef idref="S1.Ex36.m1.16"/>
                      <XMRef idref="S1.Ex36.m1.18"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                      <XMTok name="Psi" role="UNKNOWN" xml:id="S1.Ex36.m1.16">Ψ</XMTok>
                      <XMTok role="CLOSE" stretchy="true">|</XMTok>
                      <XMApp xml:id="S1.Ex36.m1.17">
                        <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok role="OPEN" stretchy="true">|</XMTok>
                      <XMTok name="Psi" role="UNKNOWN" xml:id="S1.Ex36.m1.18">Ψ</XMTok>
                      <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex37">
          <Math mode="display" tex="\matrixelement{n}{A}{m}\qquad\matrixelement{n}{A}{m}\qquad\matrixelement{n}{%&#10;\frac{X}{Y}}{m}\qquad\matrixelement*{n}{\frac{X}{Y}}{X^{Y}}\qquad%&#10;\matrixelement{n}{\frac{X}{Y}}{m}" text="list@(expectation-value@(A, n, m), expectation-value@(A, n, m), expectation-value@(X / Y, n, m), expectation-value@(X / Y, n, X ^ Y), expectation-value@(X / Y, n, m))" xml:id="S1.Ex37.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex37.m1.16"/>
                  <XMRef idref="S1.Ex37.m1.17"/>
                  <XMRef idref="S1.Ex37.m1.18"/>
                  <XMRef idref="S1.Ex37.m1.19"/>
                  <XMRef idref="S1.Ex37.m1.20"/>
                </XMApp>
                <XMWrap>
                  <XMDual xml:id="S1.Ex37.m1.16">
                    <XMApp>
                      <XMTok meaning="expectation-value"/>
                      <XMRef idref="S1.Ex37.m1.2"/>
                      <XMRef idref="S1.Ex37.m1.1"/>
                      <XMRef idref="S1.Ex37.m1.3"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex37.m1.1">n</XMTok>
                      <XMTok role="CLOSE" stretchy="true">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex37.m1.2">A</XMTok>
                      <XMTok role="OPEN" stretchy="true">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex37.m1.3">m</XMTok>
                      <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex37.m1.17">
                    <XMApp>
                      <XMTok meaning="expectation-value"/>
                      <XMRef idref="S1.Ex37.m1.5"/>
                      <XMRef idref="S1.Ex37.m1.4"/>
                      <XMRef idref="S1.Ex37.m1.6"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex37.m1.4">n</XMTok>
                      <XMTok role="CLOSE" stretchy="true">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex37.m1.5">A</XMTok>
                      <XMTok role="OPEN" stretchy="true">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex37.m1.6">m</XMTok>
                      <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex37.m1.18">
                    <XMApp>
                      <XMTok meaning="expectation-value"/>
                      <XMRef idref="S1.Ex37.m1.8"/>
                      <XMRef idref="S1.Ex37.m1.7"/>
                      <XMRef idref="S1.Ex37.m1.9"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex37.m1.7">n</XMTok>
                      <XMTok role="CLOSE" stretchy="true">|</XMTok>
                      <XMApp xml:id="S1.Ex37.m1.8">
                        <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok role="OPEN" stretchy="true">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex37.m1.9">m</XMTok>
                      <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex37.m1.19">
                    <XMApp>
                      <XMTok meaning="expectation-value"/>
                      <XMRef idref="S1.Ex37.m1.11"/>
                      <XMRef idref="S1.Ex37.m1.10"/>
                      <XMRef idref="S1.Ex37.m1.12"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="false">⟨</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex37.m1.10">n</XMTok>
                      <XMTok role="CLOSE" stretchy="false">|</XMTok>
                      <XMApp xml:id="S1.Ex37.m1.11">
                        <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok role="OPEN" stretchy="false">|</XMTok>
                      <XMApp xml:id="S1.Ex37.m1.12">
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" fontsize="70%" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok name="rangle" role="CLOSE" stretchy="false">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex37.m1.20">
                    <XMApp>
                      <XMTok meaning="expectation-value"/>
                      <XMRef idref="S1.Ex37.m1.14"/>
                      <XMRef idref="S1.Ex37.m1.13"/>
                      <XMRef idref="S1.Ex37.m1.15"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex37.m1.13">n</XMTok>
                      <XMTok role="CLOSE" stretchy="true">|</XMTok>
                      <XMApp xml:id="S1.Ex37.m1.14">
                        <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                        <XMTok font="italic" role="UNKNOWN">X</XMTok>
                        <XMTok font="italic" role="UNKNOWN">Y</XMTok>
                      </XMApp>
                      <XMTok role="OPEN" stretchy="true">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S1.Ex37.m1.15">m</XMTok>
                      <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                    </XMWrap>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
      </para>
    </subsection>
    <subsection inlist="toc" xml:id="S1.SS7">
      <tags>
        <tag>1.7</tag>
        <tag role="refnum">1.7</tag>
        <tag role="typerefnum">§1.7</tag>
      </tags>
      <title><tag close=" ">1.7</tag>Matrix macros</title>
      <para xml:id="S1.SS7.p1">
        <equation xml:id="S1.Ex38">
          <Math mode="display" tex="\begin{pmatrix}1&amp;0\\&#10;0&amp;1\\&#10;a&amp;b\end{pmatrix}" text="matrix@(Array[[1, 0], [0, 1], [a, b]])" xml:id="S1.Ex38.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="matrix"/>
                  <XMRef idref="S1.Ex38.m1.1"/>
                </XMApp>
                <XMWrap>
                  <XMTok role="OPEN" stretchy="true">(</XMTok>
                  <XMArray xml:id="S1.Ex38.m1.1">
                    <XMRow>
                      <XMCell align="center">
                        <XMTok meaning="1" role="NUMBER">1</XMTok>
                      </XMCell>
                      <XMCell align="center">
                        <XMTok meaning="0" role="NUMBER">0</XMTok>
                      </XMCell>
                    </XMRow>
                    <XMRow>
                      <XMCell align="center">
                        <XMTok meaning="0" role="NUMBER">0</XMTok>
                      </XMCell>
                      <XMCell align="center">
                        <XMTok meaning="1" role="NUMBER">1</XMTok>
                      </XMCell>
                    </XMRow>
                    <XMRow>
                      <XMCell align="center">
                        <XMTok font="italic" role="UNKNOWN">a</XMTok>
                      </XMCell>
                      <XMCell align="center">
                        <XMTok font="italic" role="UNKNOWN">b</XMTok>
                      </XMCell>
                    </XMRow>
                  </XMArray>
                  <XMTok role="CLOSE" stretchy="true">)</XMTok>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
      </para>
      <para xml:id="S1.SS7.p2">
        <equation xml:id="S1.Ex39">
          <Math mode="display" tex="\begin{pmatrix}\matrixquantity{1&amp;0\\&#10;0&amp;1}&amp;\matrixquantity{a\\&#10;b}\\&#10;\matrixquantity{c&amp;d}&amp;e\end{pmatrix}" text="matrix@(Array[[matrix@(Array[[1, 0], [0, 1]]), matrix@(Array[[a], [b]])], [matrix@(Array[[c, d]]), e]])" xml:id="S1.Ex39.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="matrix"/>
                  <XMRef idref="S1.Ex39.m1.1"/>
                </XMApp>
                <XMWrap>
                  <XMTok role="OPEN" stretchy="true">(</XMTok>
                  <XMArray xml:id="S1.Ex39.m1.1">
                    <XMRow>
                      <XMCell align="center">
                        <XMDual>
                          <XMRef idref="S1.Ex39.m1.1.1"/>
                          <XMDual xml:id="S1.Ex39.m1.1.1">
                            <XMApp>
                              <XMTok meaning="matrix"/>
                              <XMRef idref="S1.Ex39.m1.1.1.1"/>
                            </XMApp>
                            <XMArray xml:id="S1.Ex39.m1.1.1.1">
                              <XMRow>
                                <XMCell align="center">
                                  <XMTok meaning="1" role="NUMBER">1</XMTok>
                                </XMCell>
                                <XMCell align="center">
                                  <XMTok meaning="0" role="NUMBER">0</XMTok>
                                </XMCell>
                              </XMRow>
                              <XMRow>
                                <XMCell align="center">
                                  <XMTok meaning="0" role="NUMBER">0</XMTok>
                                </XMCell>
                                <XMCell align="center">
                                  <XMTok meaning="1" role="NUMBER">1</XMTok>
                                </XMCell>
                              </XMRow>
                            </XMArray>
                          </XMDual>
                        </XMDual>
                      </XMCell>
                      <XMCell align="center">
                        <XMDual>
                          <XMRef idref="S1.Ex39.m1.1.2"/>
                          <XMDual xml:id="S1.Ex39.m1.1.2">
                            <XMApp>
                              <XMTok meaning="matrix"/>
                              <XMRef idref="S1.Ex39.m1.1.2.1"/>
                            </XMApp>
                            <XMArray xml:id="S1.Ex39.m1.1.2.1">
                              <XMRow>
                                <XMCell align="center">
                                  <XMTok font="italic" role="UNKNOWN">a</XMTok>
                                </XMCell>
                              </XMRow>
                              <XMRow>
                                <XMCell align="center">
                                  <XMTok font="italic" role="UNKNOWN">b</XMTok>
                                </XMCell>
                              </XMRow>
                            </XMArray>
                          </XMDual>
                        </XMDual>
                      </XMCell>
                    </XMRow>
                    <XMRow>
                      <XMCell align="center">
                        <XMDual>
                          <XMRef idref="S1.Ex39.m1.1.3"/>
                          <XMDual xml:id="S1.Ex39.m1.1.3">
                            <XMApp>
                              <XMTok meaning="matrix"/>
                              <XMRef idref="S1.Ex39.m1.1.3.1"/>
                            </XMApp>
                            <XMArray xml:id="S1.Ex39.m1.1.3.1">
                              <XMRow>
                                <XMCell align="center">
                                  <XMTok font="italic" role="UNKNOWN">c</XMTok>
                                </XMCell>
                                <XMCell align="center">
                                  <XMTok font="italic" role="UNKNOWN">d</XMTok>
                                </XMCell>
                              </XMRow>
                            </XMArray>
                          </XMDual>
                        </XMDual>
                      </XMCell>
                      <XMCell align="center">
                        <XMTok font="italic" role="UNKNOWN">e</XMTok>
                      </XMCell>
                    </XMRow>
                  </XMArray>
                  <XMTok role="CLOSE" stretchy="true">)</XMTok>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
      </para>
      <para xml:id="S1.SS7.p3">
        <equation xml:id="S1.Ex40">
          <Math mode="display" tex="\matrixquantity(\matrixquantity{1&amp;0\\&#10;0&amp;1}&amp;\matrixquantity{a\\&#10;b}\\&#10;\matrixquantity{c&amp;d}&amp;e)" text="matrix@(Array[[matrix@(Array[[1, 0], [0, 1]]), matrix@(Array[[a], [b]])], [matrix@(Array[[c, d]]), e]])" xml:id="S1.Ex40.m1">
            <XMath>
              <XMDual>
                <XMRef idref="S1.Ex40.m1.1"/>
                <XMWrap>
                  <XMTok role="OPEN" stretchy="true">(</XMTok>
                  <XMDual xml:id="S1.Ex40.m1.1">
                    <XMApp>
                      <XMTok meaning="matrix"/>
                      <XMRef idref="S1.Ex40.m1.1.1"/>
                    </XMApp>
                    <XMArray xml:id="S1.Ex40.m1.1.1">
                      <XMRow>
                        <XMCell align="center">
                          <XMDual>
                            <XMRef idref="S1.Ex40.m1.1.1.1"/>
                            <XMDual xml:id="S1.Ex40.m1.1.1.1">
                              <XMApp>
                                <XMTok meaning="matrix"/>
                                <XMRef idref="S1.Ex40.m1.1.1.1.1"/>
                              </XMApp>
                              <XMArray xml:id="S1.Ex40.m1.1.1.1.1">
                                <XMRow>
                                  <XMCell align="center">
                                    <XMTok meaning="1" role="NUMBER">1</XMTok>
                                  </XMCell>
                                  <XMCell align="center">
                                    <XMTok meaning="0" role="NUMBER">0</XMTok>
                                  </XMCell>
                                </XMRow>
                                <XMRow>
                                  <XMCell align="center">
                                    <XMTok meaning="0" role="NUMBER">0</XMTok>
                                  </XMCell>
                                  <XMCell align="center">
                                    <XMTok meaning="1" role="NUMBER">1</XMTok>
                                  </XMCell>
                                </XMRow>
                              </XMArray>
                            </XMDual>
                          </XMDual>
                        </XMCell>
                        <XMCell align="center">
                          <XMDual>
                            <XMRef idref="S1.Ex40.m1.1.1.2"/>
                            <XMDual xml:id="S1.Ex40.m1.1.1.2">
                              <XMApp>
                                <XMTok meaning="matrix"/>
                                <XMRef idref="S1.Ex40.m1.1.1.2.1"/>
                              </XMApp>
                              <XMArray xml:id="S1.Ex40.m1.1.1.2.1">
                                <XMRow>
                                  <XMCell align="center">
                                    <XMTok font="italic" role="UNKNOWN">a</XMTok>
                                  </XMCell>
                                </XMRow>
                                <XMRow>
                                  <XMCell align="center">
                                    <XMTok font="italic" role="UNKNOWN">b</XMTok>
                                  </XMCell>
                                </XMRow>
                              </XMArray>
                            </XMDual>
                          </XMDual>
                        </XMCell>
                      </XMRow>
                      <XMRow>
                        <XMCell align="center">
                          <XMDual>
                            <XMRef idref="S1.Ex40.m1.1.1.3"/>
                            <XMDual xml:id="S1.Ex40.m1.1.1.3">
                              <XMApp>
                                <XMTok meaning="matrix"/>
                                <XMRef idref="S1.Ex40.m1.1.1.3.1"/>
                              </XMApp>
                              <XMArray xml:id="S1.Ex40.m1.1.1.3.1">
                                <XMRow>
                                  <XMCell align="center">
                                    <XMTok font="italic" role="UNKNOWN">c</XMTok>
                                  </XMCell>
                                  <XMCell align="center">
                                    <XMTok font="italic" role="UNKNOWN">d</XMTok>
                                  </XMCell>
                                </XMRow>
                              </XMArray>
                            </XMDual>
                          </XMDual>
                        </XMCell>
                        <XMCell align="center">
                          <XMTok font="italic" role="UNKNOWN">e</XMTok>
                        </XMCell>
                      </XMRow>
                    </XMArray>
                  </XMDual>
                  <XMTok role="CLOSE" stretchy="true">)</XMTok>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
      </para>
      <para xml:id="S1.SS7.p4">
        <p>But, alignment is illusion</p>
        <equation xml:id="S1.Ex41">
          <Math mode="display" tex="\begin{pmatrix}\matrixquantity{1&amp;0\\&#10;0&amp;1}&amp;\matrixquantity{\displaystyle\frac{x}{y}\\&#10;b}\\&#10;\matrixquantity{u+v+w+x+y+z&amp;d}&amp;e\end{pmatrix}" text="matrix@(Array[[matrix@(Array[[1, 0], [0, 1]]), matrix@(Array[[x / y], [b]])], [matrix@(Array[[u + v + w + x + y + z, d]]), e]])" xml:id="S1.Ex41.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="matrix"/>
                  <XMRef idref="S1.Ex41.m1.1"/>
                </XMApp>
                <XMWrap>
                  <XMTok role="OPEN" stretchy="true">(</XMTok>
                  <XMArray xml:id="S1.Ex41.m1.1">
                    <XMRow>
                      <XMCell align="center">
                        <XMDual>
                          <XMRef idref="S1.Ex41.m1.1.1"/>
                          <XMDual xml:id="S1.Ex41.m1.1.1">
                            <XMApp>
                              <XMTok meaning="matrix"/>
                              <XMRef idref="S1.Ex41.m1.1.1.1"/>
                            </XMApp>
                            <XMArray xml:id="S1.Ex41.m1.1.1.1">
                              <XMRow>
                                <XMCell align="center">
                                  <XMTok meaning="1" role="NUMBER">1</XMTok>
                                </XMCell>
                                <XMCell align="center">
                                  <XMTok meaning="0" role="NUMBER">0</XMTok>
                                </XMCell>
                              </XMRow>
                              <XMRow>
                                <XMCell align="center">
                                  <XMTok meaning="0" role="NUMBER">0</XMTok>
                                </XMCell>
                                <XMCell align="center">
                                  <XMTok meaning="1" role="NUMBER">1</XMTok>
                                </XMCell>
                              </XMRow>
                            </XMArray>
                          </XMDual>
                        </XMDual>
                      </XMCell>
                      <XMCell align="center">
                        <XMDual>
                          <XMRef idref="S1.Ex41.m1.1.2"/>
                          <XMDual xml:id="S1.Ex41.m1.1.2">
                            <XMApp>
                              <XMTok meaning="matrix"/>
                              <XMRef idref="S1.Ex41.m1.1.2.1"/>
                            </XMApp>
                            <XMArray xml:id="S1.Ex41.m1.1.2.1">
                              <XMRow>
                                <XMCell align="center">
                                  <XMApp>
                                    <XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
                                    <XMTok font="italic" role="UNKNOWN">x</XMTok>
                                    <XMTok font="italic" role="UNKNOWN">y</XMTok>
                                  </XMApp>
                                </XMCell>
                              </XMRow>
                              <XMRow>
                                <XMCell align="center">
                                  <XMTok font="italic" role="UNKNOWN">b</XMTok>
                                </XMCell>
                              </XMRow>
                            </XMArray>
                          </XMDual>
                        </XMDual>
                      </XMCell>
                    </XMRow>
                    <XMRow>
                      <XMCell align="center">
                        <XMDual>
                          <XMRef idref="S1.Ex41.m1.1.3"/>
                          <XMDual xml:id="S1.Ex41.m1.1.3">
                            <XMApp>
                              <XMTok meaning="matrix"/>
                              <XMRef idref="S1.Ex41.m1.1.3.1"/>
                            </XMApp>
                            <XMArray xml:id="S1.Ex41.m1.1.3.1">
                              <XMRow>
                                <XMCell align="center">
                                  <XMApp>
                                    <XMTok meaning="plus" role="ADDOP">+</XMTok>
                                    <XMTok font="italic" role="UNKNOWN">u</XMTok>
                                    <XMTok font="italic" role="UNKNOWN">v</XMTok>
                                    <XMTok font="italic" role="UNKNOWN">w</XMTok>
                                    <XMTok font="italic" role="UNKNOWN">x</XMTok>
                                    <XMTok font="italic" role="UNKNOWN">y</XMTok>
                                    <XMTok font="italic" role="UNKNOWN">z</XMTok>
                                  </XMApp>
                                </XMCell>
                                <XMCell align="center">
                                  <XMTok font="italic" role="UNKNOWN">d</XMTok>
                                </XMCell>
                              </XMRow>
                            </XMArray>
                          </XMDual>
                        </XMDual>
                      </XMCell>
                      <XMCell align="center">
                        <XMTok font="italic" role="UNKNOWN">e</XMTok>
                      </XMCell>
                    </XMRow>
                  </XMArray>
                  <XMTok role="CLOSE" stretchy="true">)</XMTok>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
      </para>
      <para xml:id="S1.SS7.p5">
        <equation xml:id="S1.Ex42">
          <Math mode="display" tex="\matrixquantity{a&amp;b\\&#10;c&amp;d}\qquad{\matrixquantity(a&amp;b\\&#10;c&amp;d)}\qquad{\matrixquantity*(a&amp;b\\&#10;c&amp;d)}\qquad{\matrixquantity[a&amp;b\\&#10;c&amp;d]}\qquad{\matrixquantity|a&amp;b\\&#10;c&amp;d|}\qquad\smallmatrixquantity{a&amp;b\\&#10;c&amp;d}\qquad{\matrixdeterminant{\begin{matrix}a&amp;b\\&#10;c&amp;d\end{matrix}}}\qquad{\smallmatrixdeterminant{\begin{smallmatrix}a&amp;b\\&#10;c&amp;d\end{smallmatrix}}}" text="list@(matrix@(Array[[a, b], [c, d]]), matrix@(Array[[a, b], [c, d]]), matrix@(Array[[a, b], [c, d]]), matrix@(Array[[a, b], [c, d]]), matrix@(Array[[a, b], [c, d]]), matrix@(Array[[a, b], [c, d]]), determinant@(matrix@(Array[[a, b], [c, d]])), determinant@(Array[[a, b], [c, d]]))" xml:id="S1.Ex42.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex42.m1.9"/>
                  <XMRef idref="S1.Ex42.m1.10"/>
                  <XMRef idref="S1.Ex42.m1.11"/>
                  <XMRef idref="S1.Ex42.m1.12"/>
                  <XMRef idref="S1.Ex42.m1.13"/>
                  <XMRef idref="S1.Ex42.m1.14"/>
                  <XMRef idref="S1.Ex42.m1.15"/>
                  <XMRef idref="S1.Ex42.m1.16"/>
                </XMApp>
                <XMWrap>
                  <XMDual xml:id="S1.Ex42.m1.9">
                    <XMRef idref="S1.Ex42.m1.1"/>
                    <XMDual xml:id="S1.Ex42.m1.1">
                      <XMApp>
                        <XMTok meaning="matrix"/>
                        <XMRef idref="S1.Ex42.m1.1.1"/>
                      </XMApp>
                      <XMArray xml:id="S1.Ex42.m1.1.1">
                        <XMRow>
                          <XMCell align="center">
                            <XMTok font="italic" role="UNKNOWN">a</XMTok>
                          </XMCell>
                          <XMCell align="center">
                            <XMTok font="italic" role="UNKNOWN">b</XMTok>
                          </XMCell>
                        </XMRow>
                        <XMRow>
                          <XMCell align="center">
                            <XMTok font="italic" role="UNKNOWN">c</XMTok>
                          </XMCell>
                          <XMCell align="center">
                            <XMTok font="italic" role="UNKNOWN">d</XMTok>
                          </XMCell>
                        </XMRow>
                      </XMArray>
                    </XMDual>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex42.m1.10">
                    <XMRef idref="S1.Ex42.m1.2"/>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">(</XMTok>
                      <XMDual xml:id="S1.Ex42.m1.2">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex42.m1.2.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex42.m1.2.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">a</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">b</XMTok>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">c</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">d</XMTok>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="CLOSE" stretchy="true">)</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex42.m1.11">
                    <XMRef idref="S1.Ex42.m1.3"/>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">(</XMTok>
                      <XMDual xml:id="S1.Ex42.m1.3">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex42.m1.3.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex42.m1.3.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">a</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">b</XMTok>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">c</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">d</XMTok>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="CLOSE" stretchy="true">)</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex42.m1.12">
                    <XMRef idref="S1.Ex42.m1.4"/>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">[</XMTok>
                      <XMDual xml:id="S1.Ex42.m1.4">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex42.m1.4.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex42.m1.4.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">a</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">b</XMTok>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">c</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">d</XMTok>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="CLOSE" stretchy="true">]</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex42.m1.13">
                    <XMRef idref="S1.Ex42.m1.5"/>
                    <XMWrap>
                      <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                      <XMDual xml:id="S1.Ex42.m1.5">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex42.m1.5.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex42.m1.5.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">a</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">b</XMTok>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">c</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">d</XMTok>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex42.m1.14">
                    <XMRef idref="S1.Ex42.m1.6"/>
                    <XMDual xml:id="S1.Ex42.m1.6">
                      <XMApp>
                        <XMTok meaning="matrix"/>
                        <XMRef idref="S1.Ex42.m1.6.1"/>
                      </XMApp>
                      <XMArray xml:id="S1.Ex42.m1.6.1">
                        <XMRow>
                          <XMCell align="center">
                            <XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
                          </XMCell>
                          <XMCell align="center">
                            <XMTok font="italic" fontsize="70%" role="UNKNOWN">b</XMTok>
                          </XMCell>
                        </XMRow>
                        <XMRow>
                          <XMCell align="center">
                            <XMTok font="italic" fontsize="70%" role="UNKNOWN">c</XMTok>
                          </XMCell>
                          <XMCell align="center">
                            <XMTok font="italic" fontsize="70%" role="UNKNOWN">d</XMTok>
                          </XMCell>
                        </XMRow>
                      </XMArray>
                    </XMDual>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex42.m1.15">
                    <XMApp>
                      <XMTok meaning="determinant"/>
                      <XMRef idref="S1.Ex42.m1.7"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                      <XMDual xml:id="S1.Ex42.m1.7">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex42.m1.7.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex42.m1.7.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">a</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">b</XMTok>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">c</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok font="italic" role="UNKNOWN">d</XMTok>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex42.m1.16">
                    <XMApp>
                      <XMTok meaning="determinant"/>
                      <XMRef idref="S1.Ex42.m1.8"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                      <XMArray xml:id="S1.Ex42.m1.8">
                        <XMRow>
                          <XMCell align="center">
                            <XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
                          </XMCell>
                          <XMCell align="center">
                            <XMTok font="italic" fontsize="70%" role="UNKNOWN">b</XMTok>
                          </XMCell>
                        </XMRow>
                        <XMRow>
                          <XMCell align="center">
                            <XMTok font="italic" fontsize="70%" role="UNKNOWN">c</XMTok>
                          </XMCell>
                          <XMCell align="center">
                            <XMTok font="italic" fontsize="70%" role="UNKNOWN">d</XMTok>
                          </XMCell>
                        </XMRow>
                      </XMArray>
                      <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                    </XMWrap>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex43">
          <Math mode="display" tex="\smallmatrixquantity(1&amp;0&amp;0\\&#10;0&amp;1&amp;0\\&#10;0&amp;0&amp;1)\qquad\smallmatrixquantity(1&amp;1&amp;1\\&#10;1&amp;1&amp;1)\qquad\smallmatrixquantity(a_{11}&amp;a_{12}&amp;a_{13}\\&#10;a_{21}&amp;a_{22}&amp;a_{23}\\&#10;a_{31}&amp;a_{32}&amp;a_{33})\qquad\smallmatrixquantity(a_{1}\\&#10;a_{2}\\&#10;a_{3})\qquad\smallmatrixquantity(a_{1}&amp;a_{2}&amp;a_{3})\qquad\smallmatrixquantity(%&#10;0&amp;0\\&#10;0&amp;0)" text="list@(matrix@(Array[[1, 0, 0], [0, 1, 0], [0, 0, 1]]), matrix@(Array[[1, 1, 1], [1, 1, 1]]), matrix@(Array[[a _ (list@(1, 1)), a _ (list@(1, 2)), a _ (list@(1, 3))], [a _ (list@(2, 1)), a _ (list@(2, 2)), a _ (list@(2, 3))], [a _ (list@(3, 1)), a _ (list@(3, 2)), a _ (list@(3, 3))]]), matrix@(Array[[a _ 1], [a _ 2], [a _ 3]]), matrix@(Array[[a _ 1, a _ 2, a _ 3]]), matrix@(Array[[0, 0], [0, 0]]))" xml:id="S1.Ex43.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex43.m1.7"/>
                  <XMRef idref="S1.Ex43.m1.8"/>
                  <XMRef idref="S1.Ex43.m1.9"/>
                  <XMRef idref="S1.Ex43.m1.10"/>
                  <XMRef idref="S1.Ex43.m1.11"/>
                  <XMRef idref="S1.Ex43.m1.12"/>
                </XMApp>
                <XMWrap>
                  <XMDual xml:id="S1.Ex43.m1.7">
                    <XMRef idref="S1.Ex43.m1.1"/>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">(</XMTok>
                      <XMDual xml:id="S1.Ex43.m1.1">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex43.m1.1.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex43.m1.1.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="CLOSE" stretchy="true">)</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex43.m1.8">
                    <XMRef idref="S1.Ex43.m1.2"/>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">(</XMTok>
                      <XMDual xml:id="S1.Ex43.m1.2">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex43.m1.2.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex43.m1.2.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="CLOSE" stretchy="true">)</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex43.m1.9">
                    <XMRef idref="S1.Ex43.m1.3"/>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">(</XMTok>
                      <XMDual xml:id="S1.Ex43.m1.3">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex43.m1.3.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex43.m1.3.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMApp>
                                <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
                                <XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
                                <XMDual>
                                  <XMApp>
                                    <XMTok meaning="list"/>
                                    <XMRef idref="S1.Ex43.m1.3.1.1"/>
                                    <XMRef idref="S1.Ex43.m1.3.1.2"/>
                                  </XMApp>
                                  <XMWrap>
                                    <XMTok fontsize="49%" meaning="1" role="NUMBER" xml:id="S1.Ex43.m1.3.1.1">1</XMTok>
                                    <XMTok fontsize="49%" role="PUNCT">⁣</XMTok>
                                    <XMTok fontsize="49%" meaning="1" role="NUMBER" xml:id="S1.Ex43.m1.3.1.2">1</XMTok>
                                  </XMWrap>
                                </XMDual>
                              </XMApp>
                            </XMCell>
                            <XMCell align="center">
                              <XMApp>
                                <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
                                <XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
                                <XMDual>
                                  <XMApp>
                                    <XMTok meaning="list"/>
                                    <XMRef idref="S1.Ex43.m1.3.1.3"/>
                                    <XMRef idref="S1.Ex43.m1.3.1.4"/>
                                  </XMApp>
                                  <XMWrap>
                                    <XMTok fontsize="49%" meaning="1" role="NUMBER" xml:id="S1.Ex43.m1.3.1.3">1</XMTok>
                                    <XMTok fontsize="49%" role="PUNCT">⁣</XMTok>
                                    <XMTok fontsize="49%" meaning="2" role="NUMBER" xml:id="S1.Ex43.m1.3.1.4">2</XMTok>
                                  </XMWrap>
                                </XMDual>
                              </XMApp>
                            </XMCell>
                            <XMCell align="center">
                              <XMApp>
                                <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
                                <XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
                                <XMDual>
                                  <XMApp>
                                    <XMTok meaning="list"/>
                                    <XMRef idref="S1.Ex43.m1.3.1.5"/>
                                    <XMRef idref="S1.Ex43.m1.3.1.6"/>
                                  </XMApp>
                                  <XMWrap>
                                    <XMTok fontsize="49%" meaning="1" role="NUMBER" xml:id="S1.Ex43.m1.3.1.5">1</XMTok>
                                    <XMTok fontsize="49%" role="PUNCT">⁣</XMTok>
                                    <XMTok fontsize="49%" meaning="3" role="NUMBER" xml:id="S1.Ex43.m1.3.1.6">3</XMTok>
                                  </XMWrap>
                                </XMDual>
                              </XMApp>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMApp>
                                <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
                                <XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
                                <XMDual>
                                  <XMApp>
                                    <XMTok meaning="list"/>
                                    <XMRef idref="S1.Ex43.m1.3.1.7"/>
                                    <XMRef idref="S1.Ex43.m1.3.1.8"/>
                                  </XMApp>
                                  <XMWrap>
                                    <XMTok fontsize="49%" meaning="2" role="NUMBER" xml:id="S1.Ex43.m1.3.1.7">2</XMTok>
                                    <XMTok fontsize="49%" role="PUNCT">⁣</XMTok>
                                    <XMTok fontsize="49%" meaning="1" role="NUMBER" xml:id="S1.Ex43.m1.3.1.8">1</XMTok>
                                  </XMWrap>
                                </XMDual>
                              </XMApp>
                            </XMCell>
                            <XMCell align="center">
                              <XMApp>
                                <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
                                <XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
                                <XMDual>
                                  <XMApp>
                                    <XMTok meaning="list"/>
                                    <XMRef idref="S1.Ex43.m1.3.1.9"/>
                                    <XMRef idref="S1.Ex43.m1.3.1.10"/>
                                  </XMApp>
                                  <XMWrap>
                                    <XMTok fontsize="49%" meaning="2" role="NUMBER" xml:id="S1.Ex43.m1.3.1.9">2</XMTok>
                                    <XMTok fontsize="49%" role="PUNCT">⁣</XMTok>
                                    <XMTok fontsize="49%" meaning="2" role="NUMBER" xml:id="S1.Ex43.m1.3.1.10">2</XMTok>
                                  </XMWrap>
                                </XMDual>
                              </XMApp>
                            </XMCell>
                            <XMCell align="center">
                              <XMApp>
                                <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
                                <XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
                                <XMDual>
                                  <XMApp>
                                    <XMTok meaning="list"/>
                                    <XMRef idref="S1.Ex43.m1.3.1.11"/>
                                    <XMRef idref="S1.Ex43.m1.3.1.12"/>
                                  </XMApp>
                                  <XMWrap>
                                    <XMTok fontsize="49%" meaning="2" role="NUMBER" xml:id="S1.Ex43.m1.3.1.11">2</XMTok>
                                    <XMTok fontsize="49%" role="PUNCT">⁣</XMTok>
                                    <XMTok fontsize="49%" meaning="3" role="NUMBER" xml:id="S1.Ex43.m1.3.1.12">3</XMTok>
                                  </XMWrap>
                                </XMDual>
                              </XMApp>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMApp>
                                <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
                                <XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
                                <XMDual>
                                  <XMApp>
                                    <XMTok meaning="list"/>
                                    <XMRef idref="S1.Ex43.m1.3.1.13"/>
                                    <XMRef idref="S1.Ex43.m1.3.1.14"/>
                                  </XMApp>
                                  <XMWrap>
                                    <XMTok fontsize="49%" meaning="3" role="NUMBER" xml:id="S1.Ex43.m1.3.1.13">3</XMTok>
                                    <XMTok fontsize="49%" role="PUNCT">⁣</XMTok>
                                    <XMTok fontsize="49%" meaning="1" role="NUMBER" xml:id="S1.Ex43.m1.3.1.14">1</XMTok>
                                  </XMWrap>
                                </XMDual>
                              </XMApp>
                            </XMCell>
                            <XMCell align="center">
                              <XMApp>
                                <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
                                <XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
                                <XMDual>
                                  <XMApp>
                                    <XMTok meaning="list"/>
                                    <XMRef idref="S1.Ex43.m1.3.1.15"/>
                                    <XMRef idref="S1.Ex43.m1.3.1.16"/>
                                  </XMApp>
                                  <XMWrap>
                                    <XMTok fontsize="49%" meaning="3" role="NUMBER" xml:id="S1.Ex43.m1.3.1.15">3</XMTok>
                                    <XMTok fontsize="49%" role="PUNCT">⁣</XMTok>
                                    <XMTok fontsize="49%" meaning="2" role="NUMBER" xml:id="S1.Ex43.m1.3.1.16">2</XMTok>
                                  </XMWrap>
                                </XMDual>
                              </XMApp>
                            </XMCell>
                            <XMCell align="center">
                              <XMApp>
                                <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
                                <XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
                                <XMDual>
                                  <XMApp>
                                    <XMTok meaning="list"/>
                                    <XMRef idref="S1.Ex43.m1.3.1.17"/>
                                    <XMRef idref="S1.Ex43.m1.3.1.18"/>
                                  </XMApp>
                                  <XMWrap>
                                    <XMTok fontsize="49%" meaning="3" role="NUMBER" xml:id="S1.Ex43.m1.3.1.17">3</XMTok>
                                    <XMTok fontsize="49%" role="PUNCT">⁣</XMTok>
                                    <XMTok fontsize="49%" meaning="3" role="NUMBER" xml:id="S1.Ex43.m1.3.1.18">3</XMTok>
                                  </XMWrap>
                                </XMDual>
                              </XMApp>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="CLOSE" stretchy="true">)</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex43.m1.10">
                    <XMRef idref="S1.Ex43.m1.4"/>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">(</XMTok>
                      <XMDual xml:id="S1.Ex43.m1.4">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex43.m1.4.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex43.m1.4.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMApp>
                                <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
                                <XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
                                <XMTok fontsize="49%" meaning="1" role="NUMBER">1</XMTok>
                              </XMApp>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMApp>
                                <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
                                <XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
                                <XMTok fontsize="49%" meaning="2" role="NUMBER">2</XMTok>
                              </XMApp>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMApp>
                                <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
                                <XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
                                <XMTok fontsize="49%" meaning="3" role="NUMBER">3</XMTok>
                              </XMApp>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="CLOSE" stretchy="true">)</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex43.m1.11">
                    <XMRef idref="S1.Ex43.m1.5"/>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">(</XMTok>
                      <XMDual xml:id="S1.Ex43.m1.5">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex43.m1.5.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex43.m1.5.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMApp>
                                <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
                                <XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
                                <XMTok fontsize="49%" meaning="1" role="NUMBER">1</XMTok>
                              </XMApp>
                            </XMCell>
                            <XMCell align="center">
                              <XMApp>
                                <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
                                <XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
                                <XMTok fontsize="49%" meaning="2" role="NUMBER">2</XMTok>
                              </XMApp>
                            </XMCell>
                            <XMCell align="center">
                              <XMApp>
                                <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
                                <XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
                                <XMTok fontsize="49%" meaning="3" role="NUMBER">3</XMTok>
                              </XMApp>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="CLOSE" stretchy="true">)</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex43.m1.12">
                    <XMRef idref="S1.Ex43.m1.6"/>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">(</XMTok>
                      <XMDual xml:id="S1.Ex43.m1.6">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex43.m1.6.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex43.m1.6.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="CLOSE" stretchy="true">)</XMTok>
                    </XMWrap>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex44">
          <Math mode="display" tex="\smallmatrixquantity(1&amp;0\\&#10;0&amp;1)\qquad\smallmatrixquantity(0&amp;1\\&#10;1&amp;0)\qquad\smallmatrixquantity(0&amp;-i\\&#10;i&amp;0)\qquad\smallmatrixquantity(1&amp;0\\&#10;0&amp;-1)" text="list@(matrix@(Array[[1, 0], [0, 1]]), matrix@(Array[[0, 1], [1, 0]]), matrix@(Array[[0, - imaginary-unit], [imaginary-unit, 0]]), matrix@(Array[[1, 0], [0, - 1]]))" xml:id="S1.Ex44.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex44.m1.5"/>
                  <XMRef idref="S1.Ex44.m1.6"/>
                  <XMRef idref="S1.Ex44.m1.7"/>
                  <XMRef idref="S1.Ex44.m1.8"/>
                </XMApp>
                <XMWrap>
                  <XMDual xml:id="S1.Ex44.m1.5">
                    <XMRef idref="S1.Ex44.m1.1"/>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">(</XMTok>
                      <XMDual xml:id="S1.Ex44.m1.1">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex44.m1.1.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex44.m1.1.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="CLOSE" stretchy="true">)</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex44.m1.6">
                    <XMRef idref="S1.Ex44.m1.2"/>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">(</XMTok>
                      <XMDual xml:id="S1.Ex44.m1.2">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex44.m1.2.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex44.m1.2.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="CLOSE" stretchy="true">)</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex44.m1.7">
                    <XMRef idref="S1.Ex44.m1.3"/>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">(</XMTok>
                      <XMDual xml:id="S1.Ex44.m1.3">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex44.m1.3.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex44.m1.3.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMApp>
                                <XMTok fontsize="70%" meaning="minus" role="ADDOP">-</XMTok>
                                <XMTok font="italic" fontsize="70%" meaning="imaginary-unit" name="i" role="UNKNOWN">i</XMTok>
                              </XMApp>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok font="italic" fontsize="70%" meaning="imaginary-unit" name="i" role="UNKNOWN">i</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="CLOSE" stretchy="true">)</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex44.m1.8">
                    <XMRef idref="S1.Ex44.m1.4"/>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">(</XMTok>
                      <XMDual xml:id="S1.Ex44.m1.4">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex44.m1.4.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex44.m1.4.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok fontsize="70%" meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMApp>
                                <XMTok fontsize="70%" meaning="minus" role="ADDOP">-</XMTok>
                                <XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
                              </XMApp>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="CLOSE" stretchy="true">)</XMTok>
                    </XMWrap>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex45">
          <Math mode="display" tex="\matrixquantity(1&amp;&amp;\\&#10;&amp;2&amp;\\&#10;&amp;&amp;3)\qquad\matrixquantity(1&amp;0\\&#10;0&amp;2)\qquad\matrixquantity(1&amp;\\&#10;&amp;\begin{matrix}2&amp;3\\&#10;4&amp;5\end{matrix})\qquad\matrixquantity(&amp;&amp;1\\&#10;&amp;2&amp;\\&#10;3&amp;&amp;)" text="list@(matrix@(Array[[1, absent, absent], [absent, 2, absent], [absent, absent, 3]]), matrix@(Array[[1, 0], [0, 2]]), matrix@(Array[[1, absent], [absent, matrix@(Array[[2, 3], [4, 5]])]]), matrix@(Array[[absent, absent, 1], [absent, 2, absent], [3, absent, absent]]))" xml:id="S1.Ex45.m1">
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="list"/>
                  <XMRef idref="S1.Ex45.m1.5"/>
                  <XMRef idref="S1.Ex45.m1.6"/>
                  <XMRef idref="S1.Ex45.m1.7"/>
                  <XMRef idref="S1.Ex45.m1.8"/>
                </XMApp>
                <XMWrap>
                  <XMDual xml:id="S1.Ex45.m1.5">
                    <XMRef idref="S1.Ex45.m1.1"/>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">(</XMTok>
                      <XMDual xml:id="S1.Ex45.m1.1">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex45.m1.1.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex45.m1.1.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMTok meaning="1" role="NUMBER">1</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok meaning="absent"/>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok meaning="absent"/>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok meaning="absent"/>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok meaning="2" role="NUMBER">2</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok meaning="absent"/>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok meaning="absent"/>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok meaning="absent"/>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok meaning="3" role="NUMBER">3</XMTok>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="CLOSE" stretchy="true">)</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex45.m1.6">
                    <XMRef idref="S1.Ex45.m1.2"/>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">(</XMTok>
                      <XMDual xml:id="S1.Ex45.m1.2">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex45.m1.2.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex45.m1.2.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMTok meaning="1" role="NUMBER">1</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok meaning="0" role="NUMBER">0</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok meaning="2" role="NUMBER">2</XMTok>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="CLOSE" stretchy="true">)</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex45.m1.7">
                    <XMRef idref="S1.Ex45.m1.3"/>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">(</XMTok>
                      <XMDual xml:id="S1.Ex45.m1.3">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex45.m1.3.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex45.m1.3.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMTok meaning="1" role="NUMBER">1</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok meaning="absent"/>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok meaning="absent"/>
                            </XMCell>
                            <XMCell align="center">
                              <XMDual>
                                <XMApp>
                                  <XMTok meaning="matrix"/>
                                  <XMRef idref="S1.Ex45.m1.3.1.1"/>
                                </XMApp>
                                <XMArray xml:id="S1.Ex45.m1.3.1.1">
                                  <XMRow>
                                    <XMCell align="center">
                                      <XMTok meaning="2" role="NUMBER">2</XMTok>
                                    </XMCell>
                                    <XMCell align="center">
                                      <XMTok meaning="3" role="NUMBER">3</XMTok>
                                    </XMCell>
                                  </XMRow>
                                  <XMRow>
                                    <XMCell align="center">
                                      <XMTok meaning="4" role="NUMBER">4</XMTok>
                                    </XMCell>
                                    <XMCell align="center">
                                      <XMTok meaning="5" role="NUMBER">5</XMTok>
                                    </XMCell>
                                  </XMRow>
                                </XMArray>
                              </XMDual>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="CLOSE" stretchy="true">)</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMTok font="italic" name="qquad" role="PUNCT">  </XMTok>
                  <XMDual xml:id="S1.Ex45.m1.8">
                    <XMRef idref="S1.Ex45.m1.4"/>
                    <XMWrap>
                      <XMTok role="OPEN" stretchy="true">(</XMTok>
                      <XMDual xml:id="S1.Ex45.m1.4">
                        <XMApp>
                          <XMTok meaning="matrix"/>
                          <XMRef idref="S1.Ex45.m1.4.1"/>
                        </XMApp>
                        <XMArray xml:id="S1.Ex45.m1.4.1">
                          <XMRow>
                            <XMCell align="center">
                              <XMTok meaning="absent"/>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok meaning="absent"/>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok meaning="1" role="NUMBER">1</XMTok>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok meaning="absent"/>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok meaning="2" role="NUMBER">2</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok meaning="absent"/>
                            </XMCell>
                          </XMRow>
                          <XMRow>
                            <XMCell align="center">
                              <XMTok meaning="3" role="NUMBER">3</XMTok>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok meaning="absent"/>
                            </XMCell>
                            <XMCell align="center">
                              <XMTok meaning="absent"/>
                            </XMCell>
                          </XMRow>
                        </XMArray>
                      </XMDual>
                      <XMTok role="CLOSE" stretchy="true">)</XMTok>
                    </XMWrap>
                  </XMDual>
                </XMWrap>
              </XMDual>
            </XMath>
          </Math>
        </equation>
      </para>
    </subsection>
  </section>
</document>
