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

<b>let </b><font color="Maroon">C</font> be    <a href="cat_1.html#L1" target="_self">CatStr</a> ;<br/>

<b>set </b><font color="Maroon">CP</font> = the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">CD</font> = the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">CC</font> = the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">CI</font> = the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:26"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> iff  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Olive">f</font> )
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Olive">f</font> holds <br/>(  <a href="cat_1.html#K2" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Olive">f</font> &amp;  <a href="cat_1.html#K3" target="_self">cod</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Olive">g</font> )
 <b>and </b><br/><a NAME="E3:26"/><i><font color="Green">E22</font></i>: 
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 <font color="Maroon">C</font>  st  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Olive">g</font> &amp;  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Olive">f</font> holds <br/><font color="Olive">h</font> <a href="cat_1.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Olive">g</font></span>)</span> <a href="cat_1.html#K4" target="_self">*</a> <font color="Olive">f</font>
 <b>and </b><br/><a NAME="E4:26"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> holds <br/> (  <a href="cat_1.html#K2" target="_self">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5" target="_self">id</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp;  <a href="cat_1.html#K3" target="_self">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5" target="_self">id</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; ( for <font color="Olive">f</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K5" target="_self">id</a> <font color="Olive">b</font></span>)</span> <a href="cat_1.html#K4" target="_self">*</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> ) &amp; ( for <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> holds <br/><font color="Olive">g</font> <a href="cat_1.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5" target="_self">id</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> ) )
 ;<br/>

<b>thus </b><a NAME="E5:26"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> iff the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <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" target="_self">Cod</a> of <font color="Maroon">C</font> <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" target="_self">CAT_1:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_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 the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font>, <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font>;<br/>

<a NAME="E1:26_1"/>
( the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">g</font> &amp; the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">f</font> )
 
;<br/>
<b>hence </b><a NAME="E2:26_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> iff the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <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" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T4" target="_self">Th4</a></i>;<br/>


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

<b>thus </b><a NAME="E6:26"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font>  st the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <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" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/>( the <a href="cat_1.html#U3" target="_self">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" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> &amp; the <a href="cat_1.html#U4" target="_self">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" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_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 the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font>, <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:26_2"/><i><font color="Green">E52</font></i>: 
the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <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" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:26_2"/>
( the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">g</font> &amp; the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">f</font> )
 
;<br/>
<a NAME="E3:26_2"/><b>then </b>
(  <a href="cat_1.html#K2" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">f</font> &amp;  <a href="cat_1.html#K3" target="_self">cod</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">g</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> &amp; the <a href="cat_1.html#U3" target="_self">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="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font></span>)</span> &amp; the <a href="cat_1.html#U4" target="_self">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="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i>, , <a class="txt" href="cat_1.html#E1:26"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E4:26_2"/><b>then </b>
( the <a href="cat_1.html#U3" target="_self">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" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">f</font> &amp; the <a href="cat_1.html#U4" target="_self">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" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">g</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:26_2"/>
( the <a href="cat_1.html#U3" target="_self">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" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> &amp; the <a href="cat_1.html#U4" target="_self">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" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> )
 ;<br/>


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

<b>thus </b><a NAME="E7:26"/>
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 the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font>  st the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <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" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> &amp; the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <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" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/>the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> <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" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</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" target="_self">Comp</a> of <font color="Maroon">C</font> <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" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">h</font>,<font color="Olive">g</font></span><a href="tarski.html#K4">]</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="26_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 the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font>, <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font>, <font color="Maroon">h</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:26_3"/><i><font color="Green">E53</font></i>: 
( the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <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" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> &amp; the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <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" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E2:26_3"/><i><font color="Green">E54</font></i>: 
( the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">h</font> &amp; the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">g</font> &amp; the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">g</font> &amp; the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">f</font> )
 
;<br/>
<a NAME="E3:26_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> )
 
<b>by </b><i>, <a class="txt" href="cat_1.html#E1:26"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E4:26_3"/><b>then </b><i><font color="Green">E55</font></i>: 
( <font color="Maroon">g</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">h</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:26_3"/>
( the <a href="cat_1.html#U3" target="_self">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" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> &amp; the <a href="cat_1.html#U4" target="_self">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" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="txt" href="cat_1.html#E1:26"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E6:26_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> )
 
<b>by </b><i>, <a class="txt" href="cat_1.html#E1:26"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cat_1.html#E3:26"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E7:26_3"/><b>then </b>
( the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> <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"><font color="Maroon">g</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="cat_1.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font></span>)</span> &amp; the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> <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"><font color="Maroon">h</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:26_3"/>
the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> <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" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</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" target="_self">Comp</a> of <font color="Maroon">C</font> <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" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="txt" href="cat_1.html#E1:26"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cat_1.html#E2:26"><i><font color="Green">E21</font></i></a>, <a class="txt" href="cat_1.html#E3:26"><i><font color="Green">E22</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font>;<br/>

<a NAME="E8:26"/>
(  <a href="cat_1.html#K2" target="_self">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5" target="_self">id</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K3" target="_self">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5" target="_self">id</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K5" target="_self">id</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:26"/>
( the <a href="cat_1.html#U3" target="_self">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" target="_self">Id</a> of <font color="Maroon">C</font> <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" target="_self">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" target="_self">Id</a> of <font color="Maroon">C</font> <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> )
 ;<br/>

<b>thus </b><a NAME="E10:26"/>
for <font color="Olive">f</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font>  st the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">C</font> <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" target="_self">Comp</a> of <font color="Maroon">C</font> <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#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_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 the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:26_4"/><i><font color="Green">E56</font></i>: 
the <a href="cat_1.html#U4" target="_self">Cod</a> of <font color="Maroon">C</font> <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="E2:26_4"/>
(  <a href="cat_1.html#K2" target="_self">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5" target="_self">id</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K5" target="_self">id</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:26_4"/><b>then </b>
( the <a href="cat_1.html#U3" target="_self">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" target="_self">Id</a> of <font color="Maroon">C</font> <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" target="_self">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">f</font> )
 
;<br/>
<a NAME="E4:26_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="cat_1.html#K5" target="_self">id</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> )
 
<b>by </b><i>, , <a class="txt" href="cat_1.html#E1:26"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:26_4"/>
the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> <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#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><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 the <a href="cat_1.html#U2" target="_self">Morphisms</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E11:26"/><i><font color="Green">E57</font></i>: 
the <a href="cat_1.html#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <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="E12:26"/>
(  <a href="cat_1.html#K3" target="_self">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5" target="_self">id</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K5" target="_self">id</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E13:26"/><b>then </b>
( the <a href="cat_1.html#U4" target="_self">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" target="_self">Id</a> of <font color="Maroon">C</font> <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#U3" target="_self">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">g</font> )
 
;<br/>
<a NAME="E14:26"/><b>then </b>
( <font color="Maroon">g</font> <a href="cat_1.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5" target="_self">id</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> )
 
<b>by </b><i>, , <a class="txt" href="cat_1.html#E1:26"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:26"/>
the <a href="cat_1.html#U5" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U6" target="_self">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>


</div>
