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

<b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>set </b><font color="Maroon">C</font> =  <a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font>;<br/>
<b>set </b><font color="Maroon">CP</font> = the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">CD</font> = the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">CC</font> = the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">CI</font> = the <a href="cat_1.html#U6">Id</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/>
<b>thus </b><a NAME="E1:66"/>
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> iff the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> )
  <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>, <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/>

<a NAME="E1:66_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T7">CAT_1:7</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:66_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> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> iff the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E2:66"/>
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>  st the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/>( the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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> the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> &amp; the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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> the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>, <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/>

<a NAME="E1:66_2"/>
the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T9">CAT_1:9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">gf</font> = the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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> as   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:66_2"/>
( the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp; the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp; the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">gf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp; the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">gf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T65">FUNCT_2:65</a></i>;<br/>
<b>hence </b><a NAME="E4:66_2"/>
( the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> implies ( the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> &amp; the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> ) )
 ;<br/>


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

<b>thus </b><a NAME="E3:66"/>
for <font color="Olive">f</font>, <font color="Olive">g</font>, <font color="Olive">h</font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>  st the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> &amp; the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/>the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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">the <a href="cat_1.html#U5">Comp</a> of <span class="p3">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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">the <a href="cat_1.html#U5">Comp</a> of <span class="p3">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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="66_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>, <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>, <font color="Maroon">h</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/>

<a NAME="E1:66_3"/>
( the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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> <font color="Maroon">m</font> &amp; the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><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> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T9">CAT_1:9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">gf</font> = the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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>, <font color="Maroon">hg</font> = the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><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> as   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:66_3"/>
( the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">h</font>,<font color="Maroon">gf</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">hg</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T9">CAT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E4:66_3"/>
( the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> &amp; the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> implies the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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">the <a href="cat_1.html#U5">Comp</a> of <span class="p3">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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">the <a href="cat_1.html#U5">Comp</a> of <span class="p3">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/>

<a NAME="E4:66"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:66"/>
( the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <span class="p2">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T65">FUNCT_2:65</a></i>;<br/>

<b>thus </b><a NAME="E6:66"/>
for <font color="Olive">f</font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>  st the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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/>the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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">the <a href="cat_1.html#U6">Id</a> of <span class="p3">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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="66_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/>

<a NAME="E1:66_4"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:66_4"/>
( the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> implies the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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">the <a href="cat_1.html#U6">Id</a> of <span class="p3">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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_1.html#T9">CAT_1:9</a></i>;<br/>


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

<b>let </b><font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/>

<b>assume </b><a NAME="E7:66"/>
the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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="E8:66"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:66"/>
the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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">the <a href="cat_1.html#U6">Id</a> of <span class="p3">(<span class="default"><a href="cat_4.html#K23" target="_self">c1Cat*</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</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_1.html#T9">CAT_1:9</a></i>;<br/>


</div>
