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<?xml version="1.0" encoding="UTF-8"?>
<?latexml searchpaths=".,../../sty,../../rnc,../../rnc/omdoc,../../sty/modules,../../sty/statements,../../sty/sproof,../../sty/omtext,../../sty/omdoc,../../sty/sref,../../sty/presentation,../../sty/dcm,../../sty/reqdoc,../../sty/omd,../../sty/mikoslides,../../sty/problem,../../sty/assignment,../../sty/etc,/Users/kohlhase/stc/stex/example/background,../../sty,../../rnc,../../rnc/omdoc,../../sty/modules,../../sty/statements,../../sty/sproof,../../sty/omtext,../../sty/omdoc,../../sty/sref,../../sty/presentation,../../sty/dcm,../../sty/reqdoc,../../sty/omd,../../sty/mikoslides,../../sty/problem,../../sty/assignment,../../sty/etc,/Users/kohlhase/stc/stex/example/background"?>
<?latexml class="omdoc"?>
<?latexml package="stex"?>
<?latexml package="hyperref,amssymb"?>
<?latexml RelaxNGSchema="omdoc+ltxml"?>
<omdoc xmlns="http://omdoc.org/ns" xmlns:stex="http://kwarc.info/ns/sTeX" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:om="http://www.openmath.org/OpenMath" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:omdoc="http://omdoc.org/ns" stex:srcref="all.tex#textrange(from=5;1,to=5;17)" xml:id="all.omdoc">
  <omgroup layout="sectioning" xml:id="sec.functions" about="#sec.functions" stex:srcref="functions.tex#textrange(from=1;0,to=11;13)">
    <metadata xml:id="sec.functions.p1" about="#sec.functions.p1" stex:srcref="functions.tex#textrange(from=1;0,to=10;12)">
      <dc:title stex:srcref="functions.tex#textrange(from=1;0,to=11;13)" xml:id="sec.functions.p1.p1" about="#sec.functions.p1.p1">
        <p xml:id="sec.functions.p1.p1.p1" about="#sec.functions.p1.p1.p1" stex:srcref="functions.tex#textrange(from=1;0,to=11;13)">Functions</p>
      </dc:title>
    </metadata>
    <theory xml:id="functions" about="#functions" stex:srcref="functions.tex#textrange(from=2;0,to=10;12)">
      <symbol name="cart" xml:id="cart.sym" about="#cart.sym" stex:srcref="functions.tex#textrange(from=3;0,to=3;55)"/>
      <notation cd="functions" name="cart">
        <prototype>
          <om:OMA>
            <om:OMS cd="functions" cr="fun" name="cart"/>
            <exprlist name="args">
              <expr name="arg"/>
            </exprlist>
          </om:OMA>
        </prototype>
        <rendering precedence="500">
          <m:mrow>
            <m:mo egroup="fence" fence="true">(</m:mo>
            <iterate name="args">
              <separator>
                <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\crossrefOp[fun]{\times}" xml:id="functions.m2" text="CROSSREFOP@(*)">
                  <om:OMOBJ>
                    <om:OMA>
                      <om:OMV name=""/>
                      <om:OMS cd="arith1" name="times"/>
                    </om:OMA>
                  </om:OMOBJ>
                  <m:math display="inline">
                    <m:mo cr="fun">×</m:mo>
                  </m:math>
                </Math>
              </separator>
              <render name="arg"/>
            </iterate>
            <m:mo egroup="fence" fence="true">)</m:mo>
          </m:mrow>
        </rendering>
      </notation>
      <symbol name="fun" xml:id="fun.sym" about="#fun.sym" stex:srcref="functions.tex#textrange(from=4;1,to=4;72)"/>
      <notation cd="functions" name="fun">
        <prototype>
          <om:OMA>
            <om:OMS cd="functions" cr="fun" name="fun"/>
            <expr name="arg1"/>
            <expr name="arg2"/>
            <expr name="arg3"/>
          </om:OMA>
        </prototype>
        <rendering argprec="          " precedence="200">
          <m:mrow>
            <m:mo egroup="fence" fence="true">(</m:mo>
            <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="arg:1" xml:id="functions.m5" text="arg:1">
              <om:OMOBJ>
                <om:OMV name="arg:1"/>
              </om:OMOBJ>
              <m:math display="inline">
                <m:mi>arg:1</m:mi>
              </m:math>
            </Math>
            <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\crossrefOp[fun]{\colon}" xml:id="functions.m6" text="CROSSREFOP@(colon)">
              <om:OMOBJ>
                <om:OMA>
                  <om:OMV name=""/>
                  <om:OMV name=":"/>
                </om:OMA>
              </om:OMOBJ>
              <m:math display="inline">
                <m:mo cr="fun">:</m:mo>
              </m:math>
            </Math>
            <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="arg:2" xml:id="functions.m7" text="arg:2">
              <om:OMOBJ>
                <om:OMV name="arg:2"/>
              </om:OMOBJ>
              <m:math display="inline">
                <m:mi>arg:2</m:mi>
              </m:math>
            </Math>
            <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\crossrefOp[fun]{\rightarrow}" xml:id="functions.m8" text="CROSSREFOP@(rightarrow)">
              <om:OMOBJ>
                <om:OMA>
                  <om:OMV name=""/>
                  <om:OMV name="rightarrow"/>
                </om:OMA>
              </om:OMOBJ>
              <m:math display="inline">
                <m:mo cr="fun">→</m:mo>
              </m:math>
            </Math>
            <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="arg:3" xml:id="functions.m9" text="arg:3">
              <om:OMOBJ>
                <om:OMV name="arg:3"/>
              </om:OMOBJ>
              <m:math display="inline">
                <m:mi>arg:3</m:mi>
              </m:math>
            </Math>
            <m:mo egroup="fence" fence="true">)</m:mo>
          </m:mrow>
        </rendering>
      </notation>
      <symbol name="inset" xml:id="inset.sym" about="#inset.sym" stex:srcref="functions.tex#textrange(from=5;1,to=5;46)"/>
      <notation cd="functions" name="inset">
        <prototype>
          <om:OMA>
            <om:OMS cd="functions" cr="fun" name="inset"/>
            <expr name="arg1"/>
            <expr name="arg2"/>
          </om:OMA>
        </prototype>
        <rendering argprec="          " precedence="650">
          <m:mrow>
            <m:mo egroup="fence" fence="true">(</m:mo>
            <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="arg:1" xml:id="functions.m12" text="arg:1">
              <om:OMOBJ>
                <om:OMV name="arg:1"/>
              </om:OMOBJ>
              <m:math display="inline">
                <m:mi>arg:1</m:mi>
              </m:math>
            </Math>
            <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\crossrefOp[fun]{\in}" xml:id="functions.m13" text="CROSSREFOP@(element-of)">
              <om:OMOBJ>
                <om:OMA>
                  <om:OMV name=""/>
                  <om:OMS cd="latexml" name="element-of"/>
                </om:OMA>
              </om:OMOBJ>
              <m:math display="inline">
                <m:mo cr="fun">∈</m:mo>
              </m:math>
            </Math>
            <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="arg:2" xml:id="functions.m14" text="arg:2">
              <om:OMOBJ>
                <om:OMV name="arg:2"/>
              </om:OMOBJ>
              <m:math display="inline">
                <m:mi>arg:2</m:mi>
              </m:math>
            </Math>
            <m:mo egroup="fence" fence="true">)</m:mo>
          </m:mrow>
        </rendering>
      </notation>
      <symbol name="function" xml:id="function.def.sym" about="#function.def.sym" stex:srcref="functions.tex#textrange(from=7;0,to=9;16)"/>
      <definition for="function" xml:id="functions.def" about="#functions.def" stex:srcref="functions.tex#textrange(from=7;0,to=9;16)">
        <CMP xml:id="functions.def.p1" about="#functions.def.p1" stex:srcref="functions.tex#textrange(from=7;0,to=9;16)">
          <p xml:id="functions.def.p1.p1" about="#functions.def.p1.p1" stex:srcref="functions.tex#textrange(from=7;0,to=9;16)">
A <idx xml:id="functions.def.p1.p1.p1" about="#functions.def.p1.p1.p1" stex:srcref="functions.tex#textrange(from=7;68,to=8;21)"><idt><term cd="functions" name="function" role="definiendum" xml:id="functions.def.p1.p1.p1.p1" about="#functions.def.p1.p1.p1.p1" stex:srcref="functions.tex#textrange(from=7;68,to=8;21)">function</term></idt><ide index="default"><idp><text xmlns="http://dlmf.nist.gov/LaTeXML">function</text></idp></ide></idx> <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\fun{f}{A}{B}" xml:id="functions.def.p1.p1.m1" text="fun@(f, A, B)"><om:OMOBJ><om:OMA><om:OMS cd="functions" name="fun"/><om:OMV name="f"/><om:OMV name="A"/><om:OMV name="B"/></om:OMA></om:OMOBJ><m:math display="inline"><m:mrow><m:mi mathcolor="red">fun</m:mi><m:mo>⁡</m:mo><m:mrow><m:mi>f</m:mi><m:mo>, </m:mo><m:mi>A</m:mi><m:mo>, </m:mo><m:mi>B</m:mi></m:mrow></m:mrow></m:math></Math> is a left-total, right-unique relation in <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\cart{A,B}" xml:id="functions.def.p1.p1.m2" text="cart@(list@(A, B))"><om:OMOBJ><om:OMA><om:OMS cd="functions" name="cart"/><om:OMA><om:OMS cd="latexml" name="list"/><om:OMV name="A"/><om:OMV name="B"/></om:OMA></om:OMA></om:OMOBJ><m:math display="inline"><m:mrow><m:mi mathcolor="red">cart</m:mi><m:mo>⁡</m:mo><m:mrow><m:mi>A</m:mi><m:mo>,</m:mo><m:mi>B</m:mi></m:mrow></m:mrow></m:math></Math>
</p>
        </CMP>
      </definition>
    </theory>
  </omgroup>
  <!-- %%% Local Variables: 
     %%% mode: LaTeX
     %%% TeX-master: "all"
     %%% End:-->
  <omgroup layout="sectioning" xml:id="sec.reals" about="#sec.reals" stex:srcref="reals.tex#textrange(from=1;0,to=16;13)">
    <metadata xml:id="sec.reals.p1" about="#sec.reals.p1" stex:srcref="reals.tex#textrange(from=1;0,to=15;12)">
      <dc:title stex:srcref="reals.tex#textrange(from=1;0,to=16;13)" xml:id="sec.reals.p1.p1" about="#sec.reals.p1.p1">
        <p xml:id="sec.reals.p1.p1.p1" about="#sec.reals.p1.p1.p1" stex:srcref="reals.tex#textrange(from=1;0,to=16;13)">Real Numbers</p>
      </dc:title>
    </metadata>
    <theory xml:id="reals" about="#reals" stex:srcref="reals.tex#textrange(from=2;0,to=15;12)">
      <symbol name="RealNumbers" xml:id="RealNumbers.sym" about="#RealNumbers.sym" stex:srcref="reals.tex#textrange(from=3;1,to=3;33)"/>
      <notation cd="reals" name="RealNumbers">
        <prototype>
          <om:OMS cd="reals" name="RealNumbers"/>
        </prototype>
        <rendering>
          <Math xmlns="http://dlmf.nist.gov/LaTeXML" xml:id="reals.m1" text="R">
            <om:OMOBJ>
              <om:OMV name="R"/>
            </om:OMOBJ>
            <m:math display="inline">
              <m:mi mathvariant="double-struck">R</m:mi>
            </m:math>
          </Math>
        </rendering>
      </notation>
      <symbol name="absval" xml:id="absval.sym" about="#absval.sym" stex:srcref="reals.tex#textrange(from=4;1,to=4;43)"/>
      <notation cd="reals" name="absval">
        <prototype>
          <om:OMA>
            <om:OMS cd="reals" cr="fun" name="absval"/>
            <expr name="arg1"/>
          </om:OMA>
        </prototype>
        <rendering argprec="          " precedence="2000">
          <m:mrow>
            <m:mo egroup="fence" fence="true">(</m:mo>
            <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\crossrefOp[fun]{|}" xml:id="reals.m2" text="CROSSREFOP@(|)">
              <om:OMOBJ>
                <om:OMA>
                  <om:OMV name=""/>
                  <om:OMV name="|"/>
                </om:OMA>
              </om:OMOBJ>
              <m:math display="inline">
                <m:mo cr="fun">|</m:mo>
              </m:math>
            </Math>
            <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="arg:1" xml:id="reals.m3" text="arg:1">
              <om:OMOBJ>
                <om:OMV name="arg:1"/>
              </om:OMOBJ>
              <m:math display="inline">
                <m:mi>arg:1</m:mi>
              </m:math>
            </Math>
            <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\crossrefOp[fun]{|}" xml:id="reals.m4" text="CROSSREFOP@(|)">
              <om:OMOBJ>
                <om:OMA>
                  <om:OMV name=""/>
                  <om:OMV name="|"/>
                </om:OMA>
              </om:OMOBJ>
              <m:math display="inline">
                <m:mo cr="fun">|</m:mo>
              </m:math>
            </Math>
            <m:mo egroup="fence" fence="true">)</m:mo>
          </m:mrow>
        </rendering>
      </notation>
      <symbol name="rfrac" xml:id="rfrac.sym" about="#rfrac.sym" stex:srcref="reals.tex#textrange(from=5;1,to=5;33)"/>
      <notation cd="reals" name="rfrac">
        <prototype>
          <om:OMA>
            <om:OMS cd="reals" cr="fun" name="rfrac"/>
            <expr name="arg1"/>
            <expr name="arg2"/>
          </om:OMA>
        </prototype>
        <rendering>
          <Math xmlns="http://dlmf.nist.gov/LaTeXML" xml:id="reals.m5" text="arg:1 / arg:2">
            <om:OMOBJ>
              <om:OMA>
                <om:OMS cd="latexml" name="divide"/>
                <om:OMV name="arg:1"/>
                <om:OMV name="arg:2"/>
              </om:OMA>
            </om:OMOBJ>
            <m:math display="inline">
              <m:mfrac>
                <m:mi>arg:1</m:mi>
                <m:mi>arg:2</m:mi>
              </m:mfrac>
            </m:math>
          </Math>
        </rendering>
      </notation>
      <symbol name="real-numbers" xml:id="real-numbers.def.sym" about="#real-numbers.def.sym" stex:srcref="reals.tex#textrange(from=7;0,to=10;16)"/>
      <definition for="real-numbers" xml:id="reals.def" about="#reals.def" stex:srcref="reals.tex#textrange(from=7;0,to=10;16)">
        <CMP xml:id="reals.def.p1" about="#reals.def.p1" stex:srcref="reals.tex#textrange(from=7;0,to=10;16)">
          <p xml:id="reals.def.p1.p1" about="#reals.def.p1.p1" stex:srcref="reals.tex#textrange(from=7;0,to=10;16)">
We denote the set of <idx xml:id="reals.def.p1.p1.p1" about="#reals.def.p1.p1.p1" stex:srcref="reals.tex#textrange(from=7;27,to=8;47)"><idt><term cd="reals" name="real-numbers" role="definiendum" xml:id="reals.def.p1.p1.p1.p1" about="#reals.def.p1.p1.p1.p1" stex:srcref="reals.tex#textrange(from=7;27,to=8;47)">real numbers</term></idt><ide index="default"><idp><text xmlns="http://dlmf.nist.gov/LaTeXML">real</text></idp><idp><text xmlns="http://dlmf.nist.gov/LaTeXML">numbers</text></idp></ide></idx> we all know and love with
<Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\RealNumbers" xml:id="reals.def.p1.p1.m1" text="RealNumbers"><om:OMOBJ><om:OMS cd="reals" name="RealNumbers"/></om:OMOBJ><m:math display="inline"><m:mi mathcolor="red">RealNumbers</m:mi></m:math></Math>.
</p>
        </CMP>
      </definition>
      <definition xml:id="absval.def" about="#absval.def" stex:srcref="reals.tex#textrange(from=12;0,to=14;16)">
        <CMP xml:id="absval.def.p1" about="#absval.def.p1" stex:srcref="reals.tex#textrange(from=12;0,to=14;16)">
          <p xml:id="absval.def.p1.p1" about="#absval.def.p1.p1" stex:srcref="reals.tex#textrange(from=12;0,to=14;16)">
The absolute value <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="\absval{r}" xml:id="absval.def.p1.p1.m1" text="absval@(r)"><om:OMOBJ><om:OMA><om:OMS cd="reals" name="absval"/><om:OMV name="r"/></om:OMA></om:OMOBJ><m:math display="inline"><m:mrow><m:mi mathcolor="red">absval</m:mi><m:mo>⁡</m:mo><m:mi>r</m:mi></m:mrow></m:math></Math> of a real number <Math xmlns="http://dlmf.nist.gov/LaTeXML" mode="inline" tex="r" xml:id="absval.def.p1.p1.m2" text="r"><om:OMOBJ><om:OMV name="r"/></om:OMOBJ><m:math display="inline"><m:mi>r</m:mi></m:math></Math>.
</p>
        </CMP>
      </definition>
    </theory>
  </omgroup>
  <!-- %%% Local Variables: 
     %%% mode: LaTeX
     %%% TeX-master: "all"
     %%% End:-->
</omdoc>