<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon" title="c3">O</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c4">M</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c5">d</font> = <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c6">c</font> = <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c7">p</font> = <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span>;<br/>
<b>set </b><font color="Maroon" title="c8">i</font> = <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span>;<br/>
<a NAME="E1:37_1_1"/><i><font color="Green" title="E19">A1</font></i>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  st <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/>( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c10">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<a NAME="E1:37_1_1_1"/><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<a NAME="E2:37_1_1_1"/><i><font color="Green" title="E21">A3</font></i>: 
( <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_3.html#T96" title="FUNCT_3:th.96">FUNCT_3:96</a></i>;<br/>
<b>assume </b><a NAME="E3:37_1_1_1"/>
<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font>
 ;<br/>

<a NAME="E4:37_1_1_1"/><b>then </b>
( the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E1:37_1_1_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="cat_2.html#E2:37_1_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E5:37_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:37_1_1"/><i><font color="Green" title="E20">A4</font></i>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  st <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> holds <br/>( the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c10">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<a NAME="E1:37_1_1_2"/><i><font color="Green" title="E21">A5</font></i>: 
( <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<b>assume </b><a NAME="E2:37_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span>
 ;<br/>

<a NAME="E3:37_1_1_2"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E1:37_1_1_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="funct_4.html#T57" title="FUNCT_4:th.57">FUNCT_4:57</a></i>;<br/>
<b>hence </b><a NAME="E4:37_1_1_2"/>
( the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:37_1_1"/><i><font color="Green" title="E21">A6</font></i>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> iff <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c10">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<a NAME="E1:37_1_1_3"/><i><font color="Green" title="E22">A7</font></i>: 
( <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<b>thus </b><a NAME="E2:37_1_1_3"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> implies <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1_1_3_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span>
 ;<br/>

<a NAME="E2:37_1_1_3_1"/><b>then </b>
( the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> &amp; <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E2:37_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="funct_3.html#T96" title="FUNCT_3:th.96">FUNCT_3:96</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1_1_3_1"/>
<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="txt" href="cat_2.html#E1:37_1_1_3"><i><font color="Green" title="E22">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:37_1_1_3"/>
<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font>
 ;<br/>

<a NAME="E4:37_1_1_3"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E1:37_1_1"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:37_1_1_3"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span>
 <b>by </b><i><a class="txt" href="cat_2.html#E1:37_1_1_3"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="funct_4.html#T57" title="FUNCT_4:th.57">FUNCT_4:57</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:37_1_1"/><i><font color="Green" title="E22">A8</font></i>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  st <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c10">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<b>assume </b><a NAME="E1:37_1_1_4"/>
<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font>
 ;<br/>

<a NAME="E2:37_1_1_4"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> &amp; <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E3:37_1_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1_1_4"/>
<span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="funct_4.html#T58" title="FUNCT_4:th.58">FUNCT_4:58</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:37_1_1"/><i><font color="Green" title="E23">A9</font></i>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  st <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/>( <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> &amp; <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c10">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<a NAME="E1:37_1_1_5"/><i><font color="Green" title="E24">A10</font></i>: 
( <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<a NAME="E2:37_1_1_5"/><i><font color="Green" title="E25">A11</font></i>: 
( <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_3.html#T96" title="FUNCT_3:th.96">FUNCT_3:96</a></i>;<br/>
<b>assume </b><a NAME="E3:37_1_1_5"/><i><font color="Green" title="E26">A12</font></i>: 
<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font>
 ;<br/>

<a NAME="E4:37_1_1_5"/><b>then </b><i><font color="Green" title="E27">A13</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span>
 
<b>by </b><i><a class="txt" href="cat_2.html#E3:37_1_1"><i><font color="Green" title="E21">A6</font></i></a></i>;<br/>
<a NAME="E5:37_1_1_5"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E1:37_1_1"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="cat_2.html#E3:37_1_1_5"><i><font color="Green" title="E26">A12</font></i></a></i>;<br/>
<a NAME="E6:37_1_1_5"/><b>then </b>
( the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> &amp; the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font> &amp; the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E2:37_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="cat_2.html#E4:37_1_1_5"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/>
<a NAME="E7:37_1_1_5"/><b>then </b><i><font color="Green" title="E28">A14</font></i>: 
( the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> &amp; the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> &amp; the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> &amp; the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> &amp; the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> &amp; the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> &amp; <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E4:37_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="cat_2.html#E3:37_1_1_5"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> = 
<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cat_2.html#E7:37_1_1_5"><i><font color="Green" title="E28">A14</font></i></a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font>
<b>by </b><i><a class="txt" href="cat_2.html#E1:37_1_1_5"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="cat_2.html#E2:37_1_1_5"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a>, <a class="txt" href="cat_2.html#E7:37_1_1_5"><i><font color="Green" title="E28">A14</font></i></a></i>
;<br/>

<b>thus </b><span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> = 
<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cat_2.html#E7:37_1_1_5"><i><font color="Green" title="E28">A14</font></i></a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font>
<b>by </b><i><a class="txt" href="cat_2.html#E1:37_1_1_5"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="cat_2.html#E2:37_1_1_5"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a>, <a class="txt" href="cat_2.html#E7:37_1_1_5"><i><font color="Green" title="E28">A14</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:37_1_1"/><i><font color="Green" title="E24">A15</font></i>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  st <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> &amp; <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">h</font>,<span class="p2">(<span class="default"><span class="p3"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">h</font>,<font color="Olive" title="b2">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<font color="Olive" title="b1">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c10">g</font>, <font color="Maroon" title="c11">h</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<b>assume </b><a NAME="E1:37_1_1_6"/><i><font color="Green" title="E25">A16</font></i>: 
( <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font> &amp; <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font> )
 ;<br/>

<a NAME="E2:37_1_1_6"/><b>then </b><i><font color="Green" title="E26">A17</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E1:37_1_1"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>
<a NAME="E3:37_1_1_6"/><b>then </b>
( the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font> &amp; the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font> &amp; the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font> &amp; the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">pgf</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c13">phg</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:37_1_1_6"/><i><font color="Green" title="E27">A18</font></i>: 
( <font color="Maroon" title="c12">pgf</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c12">pgf</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c13">phg</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c13">phg</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">pgf1</font> = <font color="Maroon" title="c12">pgf</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> ;<br/>
<b>set </b><font color="Maroon" title="c15">phg1</font> = <font color="Maroon" title="c13">phg</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> ;<br/>
<b>set </b><font color="Maroon" title="c16">pgf2</font> = <font color="Maroon" title="c12">pgf</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> ;<br/>
<b>set </b><font color="Maroon" title="c17">phg2</font> = <font color="Maroon" title="c13">phg</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> ;<br/>
<a NAME="E6:37_1_1_6"/><i><font color="Green" title="E28">A19</font></i>: 
( the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> &amp; the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E2:37_1_1_6"><i><font color="Green" title="E26">A17</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>;<br/>
<a NAME="E7:37_1_1_6"/><i><font color="Green" title="E29">A20</font></i>: 
<span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="cat_2.html#E1:37_1_1_6"><i><font color="Green" title="E25">A16</font></i></a>, <a class="txt" href="cat_2.html#E4:37_1_1"><i><font color="Green" title="E22">A8</font></i></a></i>;<br/>
<a NAME="E8:37_1_1_6"/><i><font color="Green" title="E30">A21</font></i>: 
<span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">h</font>,<font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="cat_2.html#E4:37_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="cat_2.html#E1:37_1_1_6"><i><font color="Green" title="E25">A16</font></i></a></i>;<br/>
<a NAME="E9:37_1_1_6"/>
( <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span>)</span> &amp; <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">h</font>,<font color="Maroon" title="c10">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E5:37_1_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="cat_2.html#E1:37_1_1_6"><i><font color="Green" title="E25">A16</font></i></a></i>;<br/>
<a NAME="E10:37_1_1_6"/><b>then </b>
( <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">h</font>,<span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">pgf</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">pgf</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">h</font>,<font color="Maroon" title="c10">g</font></span>)</span>,<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">phg</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">phg</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">pgf</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">phg</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">pgf</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">phg</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E4:37_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="cat_2.html#E5:37_1_1_6"><i><font color="Green" title="E27">A18</font></i></a>, <a class="txt" href="cat_2.html#E6:37_1_1_6"><i><font color="Green" title="E28">A19</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a>, <a class="txt" href="cat_2.html#E7:37_1_1_6"><i><font color="Green" title="E29">A20</font></i></a>, <a class="txt" href="cat_2.html#E8:37_1_1_6"><i><font color="Green" title="E30">A21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:37_1_1_6"/>
<span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">h</font>,<span class="p2">(<span class="default"><span class="p3"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">h</font>,<font color="Maroon" title="c10">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<font color="Maroon" title="c9">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/><a NAME="E1:37_1_1_7"/><i><font color="Green" title="E25">A22</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/><a NAME="E2:37_1_1_7"/><i><font color="Green" title="E26">A23</font></i>: 
( the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 <b>by </b><i><a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a>, <a class="ref" href="funct_3.html#T96" title="FUNCT_3:th.96">FUNCT_3:96</a></i>;<br/><b>thus </b><i><font color="Green" title="E27">A24</font></i>: <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span> = 
<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cat_2.html#E1:37_1_1_7"><i><font color="Green" title="E25">A22</font></i></a>, <a class="txt" href="cat_2.html#E2:37_1_1_7"><i><font color="Green" title="E26">A23</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">b</font>
<b>by </b><i><a class="txt" href="cat_2.html#E2:37_1_1_7"><i><font color="Green" title="E26">A23</font></i></a>, <a class="txt" href="cat_2.html#E1:37_1_1_7"><i><font color="Green" title="E25">A22</font></i></a>, <a class="ref" href="funct_3.html#T96" title="FUNCT_3:th.96">FUNCT_3:96</a></i>
;<br/><b>thus </b><i><font color="Green" title="E28">A25</font></i>: <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span> = 
<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cat_2.html#E1:37_1_1_7"><i><font color="Green" title="E25">A22</font></i></a>, <a class="txt" href="cat_2.html#E2:37_1_1_7"><i><font color="Green" title="E26">A23</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">b</font>
<b>by </b><i><a class="txt" href="cat_2.html#E2:37_1_1_7"><i><font color="Green" title="E26">A23</font></i></a>, <a class="txt" href="cat_2.html#E1:37_1_1_7"><i><font color="Green" title="E25">A22</font></i></a>, <a class="ref" href="funct_3.html#T96" title="FUNCT_3:th.96">FUNCT_3:96</a></i>
;<br/><a NAME="E5:37_1_1_7"/><i><font color="Green" title="E29">A26</font></i>: 
( <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="cat_2.html#E2:37_1_1_7"><i><font color="Green" title="E26">A23</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><b>thus </b><a NAME="E6:37_1_1_7"/>
for <font color="Olive" title="b1">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  st <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> holds <br/><span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span>,<font color="Olive" title="b1">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_7_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">f</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<a NAME="E1:37_1_1_7_3"/><i><font color="Green" title="E30">A27</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<b>assume </b><a NAME="E2:37_1_1_7_3"/><i><font color="Green" title="E31">A28</font></i>: 
<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font>
 ;<br/>

<a NAME="E3:37_1_1_7_3"/>
<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#T96" title="FUNCT_3:th.96">FUNCT_3:96</a></i>;<br/>
<a NAME="E4:37_1_1_7_3"/><b>then </b>
( the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  )
 
<b>by </b><i><a class="txt" href="cat_2.html#E1:37_1_1_7"><i><font color="Green" title="E25">A22</font></i></a>, <a class="txt" href="cat_2.html#E2:37_1_1_7_3"><i><font color="Green" title="E31">A28</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a>, <a class="txt" href="cat_2.html#E1:37_1_1_7_3"><i><font color="Green" title="E30">A27</font></i></a></i>;<br/>
<a NAME="E5:37_1_1_7_3"/><b>then </b><i><font color="Green" title="E32">A29</font></i>: 
( <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span>,<font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  )
 
<b>by </b><i><a class="txt" href="cat_2.html#E4:37_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="cat_2.html#E3:37_1_1_7"><i><font color="Green" title="E27">A24</font></i></a>, <a class="txt" href="cat_2.html#E2:37_1_1_7_3"><i><font color="Green" title="E31">A28</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a>, <a class="txt" href="cat_2.html#E1:37_1_1_7"><i><font color="Green" title="E25">A22</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:37_1_1_7_3"/>
<span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span>,<font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font>
 <b>by </b><i><a class="txt" href="cat_2.html#E1:37_1_1_7"><i><font color="Green" title="E25">A22</font></i></a>, <a class="txt" href="cat_2.html#E5:37_1_1_7"><i><font color="Green" title="E29">A26</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a>, <a class="txt" href="cat_2.html#E5:37_1_1_7_3"><i><font color="Green" title="E32">A29</font></i></a></i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon" title="c10">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/><a NAME="E7:37_1_1_7"/><i><font color="Green" title="E30">A30</font></i>: 
<font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/><b>assume </b><a NAME="E8:37_1_1_7"/><i><font color="Green" title="E31">A31</font></i>: 
<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font>
 ;<br/><a NAME="E9:37_1_1_7"/>
<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="ref" href="funct_3.html#T96" title="FUNCT_3:th.96">FUNCT_3:96</a></i>;<br/><a NAME="E10:37_1_1_7"/><b>then </b>
( the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  )
 <b>by </b><i><a class="txt" href="cat_2.html#E1:37_1_1_7"><i><font color="Green" title="E25">A22</font></i></a>, <a class="txt" href="cat_2.html#E8:37_1_1_7"><i><font color="Green" title="E31">A31</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a>, <a class="txt" href="cat_2.html#E7:37_1_1_7"><i><font color="Green" title="E30">A30</font></i></a></i>;<br/><a NAME="E11:37_1_1_7"/><b>then </b><i><font color="Green" title="E32">A32</font></i>: 
( <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">g</font>,<span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  )
 <b>by </b><i><a class="txt" href="cat_2.html#E4:37_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="cat_2.html#E4:37_1_1_7"><i><font color="Green" title="E28">A25</font></i></a>, <a class="txt" href="cat_2.html#E8:37_1_1_7"><i><font color="Green" title="E31">A31</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a>, <a class="txt" href="cat_2.html#E1:37_1_1_7"><i><font color="Green" title="E25">A22</font></i></a></i>;<br/><b>thus </b><a NAME="E12:37_1_1_7"/>
<span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<span class="p2">(<span class="default"><span class="p3"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font>
 <b>by </b><i><a class="txt" href="cat_2.html#E1:37_1_1_7"><i><font color="Green" title="E25">A22</font></i></a>, <a class="txt" href="cat_2.html#E5:37_1_1_7"><i><font color="Green" title="E29">A26</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a>, <a class="txt" href="cat_2.html#E11:37_1_1_7"><i><font color="Green" title="E32">A32</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:37_1_1"/>
 <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span>,<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span>,<span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span>,<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c2">D</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> #) is   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>
 <b>by </b><i><a class="txt" href="cat_2.html#E3:37_1_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="cat_2.html#E5:37_1_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a>, <a class="txt" href="cat_2.html#E6:37_1_1"><i><font color="Green" title="E24">A15</font></i></a></i>;<br/>


</div>
