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

<b>set </b><font color="Maroon">O</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">M</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">d</font> = <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span>;<br/>
<b>set </b><font color="Maroon">c</font> = <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span>;<br/>
<b>set </b><font color="Maroon">p</font> = <span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span>;<br/>
<b>set </b><font color="Maroon">i</font> = <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>now </b></a><div class="add"><a NAME="E1:37_1_1_1"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>  st <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Olive">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Olive">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<a NAME="E1:37_1_1_1_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E2:37_1_1_1_1"/><i><font color="Green">E21</font></i>: 
( <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_3.html#T96">FUNCT_3:96</a></i>;<br/>
<b>assume </b><a NAME="E3:37_1_1_1_1"/>
<span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E4:37_1_1_1_1"/><b>then </b>
( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span> )
 
<b>by </b><i>, <a class="txt" href="cat_2.html#E2:37_1_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E5:37_1_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E2:37_1_1_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>  st <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> holds <br/>( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<a NAME="E1:37_1_1_1_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<b>assume </b><a NAME="E2:37_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span>
 ;<br/>

<a NAME="E3:37_1_1_1_2"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> )
 
<b>by </b><i>, <a class="ref" href="funct_4.html#T57">FUNCT_4:57</a></i>;<br/>
<b>hence </b><a NAME="E4:37_1_1_1_2"/>
( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:37_1_1_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> iff <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<a NAME="E1:37_1_1_1_3"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<b>thus </b><a NAME="E2:37_1_1_1_3"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> implies <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1_1_1_3_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span>
 ;<br/>

<a NAME="E2:37_1_1_1_3_1"/><b>then </b>
( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span> &amp; <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i>, <a class="ref" href="funct_3.html#T96">FUNCT_3:96</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1_1_1_3_1"/>
<span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E3:37_1_1_1_3"/>
<span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E4:37_1_1_1_3"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:37_1_1_1_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span>
 <b>by </b><i>, <a class="ref" href="funct_4.html#T57">FUNCT_4:57</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:37_1_1_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>  st <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/><span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Olive">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Olive">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Olive">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<b>assume </b><a NAME="E1:37_1_1_1_4"/>
<span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:37_1_1_1_4"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1_1_1_4"/>
<span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="funct_4.html#T58">FUNCT_4:58</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E5:37_1_1_1"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>  st <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/>( <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> &amp; <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<a NAME="E1:37_1_1_1_5"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E2:37_1_1_1_5"/><i><font color="Green">E70</font></i>: 
( <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_3.html#T96">FUNCT_3:96</a></i>;<br/>
<b>assume </b><a NAME="E3:37_1_1_1_5"/><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E4:37_1_1_1_5"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:37_1_1_1_5"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> )
 
<b>by </b><i>, <a class="ref" href="cat_2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E6:37_1_1_1_5"/><b>then </b>
( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font> )
 
<b>by </b><i>, <a class="ref" href="cat_2.html#T3" target="_self">Th3</a>, <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>
<a NAME="E7:37_1_1_1_5"/><b>then </b><i><font color="Green">E73</font></i>: 
( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="binop_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span> &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="binop_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> &amp; <span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">g</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="binop_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i>, <a class="ref" href="cat_2.html#T2" target="_self">Th2</a>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> = 
<span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="binop_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_2.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i>, <a class="txt" href="cat_2.html#E3:37_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a>, <a class="ref" href="cat_2.html#T4" target="_self">Th4</a></i>
;<br/>

<b>thus </b><span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> = 
<span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="binop_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_2.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font>
<b>by </b><i>, <a class="txt" href="cat_2.html#E3:37_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a>, <a class="ref" href="cat_2.html#T4" target="_self">Th4</a></i>
;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E6:37_1_1_1"/>
for <font color="Olive">f</font>, <font color="Olive">g</font>, <font color="Olive">h</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>  st <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> &amp; <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/><span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">h</font>,<span class="p2">(<span class="default"><span class="p3"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">h</font>,<font color="Olive">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <font color="Maroon">h</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<b>assume </b><a NAME="E1:37_1_1_1_6"/><i><font color="Green">E74</font></i>: 
( <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> &amp; <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E2:37_1_1_1_6"/><b>then </b><i><font color="Green">E75</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:37_1_1_1_6"/><b>then </b>
( the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">pgf</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>, <font color="Maroon">phg</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:37_1_1_1_6"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">pgf</font> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">pgf</font> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">phg</font> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">phg</font> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>set </b><font color="Maroon">pgf1</font> = <font color="Maroon">pgf</font> <a href="domain_1.html#K2">`1</a> ;<br/>
<b>set </b><font color="Maroon">phg1</font> = <font color="Maroon">phg</font> <a href="domain_1.html#K2">`1</a> ;<br/>
<b>set </b><font color="Maroon">pgf2</font> = <font color="Maroon">pgf</font> <a href="domain_1.html#K3">`2</a> ;<br/>
<b>set </b><font color="Maroon">phg2</font> = <font color="Maroon">phg</font> <a href="domain_1.html#K3">`2</a> ;<br/>
<a NAME="E6:37_1_1_1_6"/><i><font color="Green">E77</font></i>: 
( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E3:37_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/>
<a NAME="E7:37_1_1_1_6"/>
( <span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">g</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pgf</font> &amp; <span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">h</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">phg</font> &amp; <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">g</font>,<font color="Maroon">f</font></span>)</span> &amp; <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">h</font>,<font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="txt" href="cat_2.html#E2:37_1_1_1"><i><font color="Green">E23</font></i></a>, </i>;<br/>
<a NAME="E8:37_1_1_1_6"/><b>then </b>
( <span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<span class="p2">(<span class="default"><span class="p3"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">pgf</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">pgf</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">h</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">phg</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">phg</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">pgf</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">phg</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">pgf</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">phg</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i>, <a class="ref" href="cat_2.html#T2" target="_self">Th2</a>, <a class="ref" href="cat_2.html#T3" target="_self">Th3</a>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E9:37_1_1_1_6"/>
<span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<span class="p2">(<span class="default"><span class="p3"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">h</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><a NAME="E7:37_1_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/><a NAME="E8:37_1_1_1"/><i><font color="Green">E79</font></i>: 
( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a>  &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a>  &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a>  &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a>  &amp; <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 <b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a>, <a class="ref" href="funct_3.html#T96">FUNCT_3:96</a></i>;<br/><b>thus </b><i><font color="Green">E80</font></i>: <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="cat_2.html#E3:37_1_1_1"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="cat_2.html#E3:37_1_1_1"><i><font color="Green">E25</font></i></a>, , <a class="ref" href="funct_3.html#T96">FUNCT_3:96</a></i>
;<br/><b>thus </b><i><font color="Green">E81</font></i>: <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="cat_2.html#E3:37_1_1_1"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="cat_2.html#E3:37_1_1_1"><i><font color="Green">E25</font></i></a>, , <a class="ref" href="funct_3.html#T96">FUNCT_3:96</a></i>
;<br/><a NAME="E11:37_1_1_1"/><i><font color="Green">E83</font></i>: 
( <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="cat_2.html#E3:37_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><b>thus </b><a NAME="E12:37_1_1_1"/>
for <font color="Olive">f</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>  st <span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> holds <br/><span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<a NAME="E1:37_1_1_1_9"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<b>assume </b><a NAME="E2:37_1_1_1_9"/><i><font color="Green">E85</font></i>: 
<span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E3:37_1_1_1_9"/>
<span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#T96">FUNCT_3:96</a></i>;<br/>
<a NAME="E4:37_1_1_1_9"/><b>then </b>
( the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a>  &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a>  )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a>, </i>;<br/>
<a NAME="E5:37_1_1_1_9"/><b>then </b><i><font color="Green">E86</font></i>: 
( <span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="binop_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="domain_1.html#K2">`1</a>  &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="domain_1.html#K3">`2</a>  )
 
<b>by </b><i>, <a class="ref" href="cat_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a>, </i>;<br/>
<b>thus </b><a NAME="E6:37_1_1_1_9"/>
<span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="cat_2.html#T4" target="_self">Th4</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a>, </i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><a NAME="E13:37_1_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/><b>assume </b><a NAME="E14:37_1_1_1"/><i><font color="Green">E88</font></i>: 
<span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E15:37_1_1_1"/>
<span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="funct_3.html#T96">FUNCT_3:96</a></i>;<br/><a NAME="E16:37_1_1_1"/><b>then </b>
( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a>  &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a>  )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a>, </i>;<br/><a NAME="E17:37_1_1_1"/><b>then </b><i><font color="Green">E89</font></i>: 
( <span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`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" target="_self">[:</a><span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="binop_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`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" target="_self">[:</a><span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="domain_1.html#K2">`1</a>  &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="domain_1.html#K3">`2</a>  )
 <b>by </b><i>, <a class="ref" href="cat_2.html#T3" target="_self">Th3</a>, , <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a>, </i>;<br/><b>thus </b><a NAME="E18:37_1_1_1"/>
<span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<span class="p2">(<span class="default"><span class="p3"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="cat_2.html#T4" target="_self">Th4</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:37_1_1"/>
 <a href="cat_1.html#G1">CatStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span>,<span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span>,<span class="p1"><a href="cat_2.html#K10" target="_self">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K10" target="_self">:|</a></span>,<span class="p1"><a href="cat_2.html#K9" target="_self">[:</a><span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">D</font></span><a href="cat_2.html#K9" target="_self">:]</a></span> #) is   <a href="cat_1.html#NM4">Category</a>
 <b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/>


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