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

<b>let </b><font color="Maroon">A</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">O</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">M</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">DOM</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">M</font>,<font color="Maroon">O</font>, <font color="Maroon">COD</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">M</font>,<font color="Maroon">O</font>;<br/>







<b>assume </b><a NAME="E1:12"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">DOM</font> <a href="funct_2.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">A</font> <a href="nattra_1.html#K1" target="_self">|</a> <font color="Maroon">M</font> &amp; <font color="Maroon">COD</font> <a href="funct_2.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">A</font> <a href="nattra_1.html#K1" target="_self">|</a> <font color="Maroon">M</font> )
 ;<br/>

<b>let </b><font color="Maroon">COMP</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">M</font>;<br/>

<b>assume </b><a NAME="E2:12"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">COMP</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">A</font> <a href="realset1.html#K1">||</a> <font color="Maroon">M</font>
 ;<br/>

<b>let </b><font color="Maroon">ID</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">O</font>,<font color="Maroon">M</font>;<br/>

<b>assume </b><a NAME="E3:12"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">ID</font> <a href="funct_2.html#R1">=</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">A</font> <a href="nattra_1.html#K1" target="_self">|</a> <font color="Maroon">O</font>
 ;<br/>

<b>set </b><font color="Maroon">C</font> =  <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #);<br/>
<a NAME="E4:12"/><i><font color="Green">E26</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/>
<div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #);<br/><b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E2:12_1"/>
 <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #)
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:12_1"/>
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="nattra_1.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E4:12_1"/>
 <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #)
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E5:12_1"/>
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="nattra_1.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:12"/><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #),the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #)</span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #), <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #);<br/><b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font>, <font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>assume </b><a NAME="E2:12_2"/>
the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
 ;<br/><b>then </b>the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g'</font> = 
the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i/>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f'</font>
<b>by </b><i/>
;<br/><a NAME="E4:12_2"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g'</font>,<font color="Maroon">f'</font></span><a href="cat_2.html#K13">]</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">A</font>
 <b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/><a NAME="E5:12_2"/>
<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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #),the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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/><b>hence </b><a NAME="E6:12_2"/>
<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">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #)
 <b>by </b><i><a class="ref" href="nattra_1.html#T1" target="_self">Th1</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:12"/>
 <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) is <a href="cat_1.html#V2">Category-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:12_3"/>
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) iff the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">CAT_1:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_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 <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #), <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #);<br/>

<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font>, <font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><a NAME="E2:12_3_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">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) implies the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_3_1_1"/><i><font color="Green">E37</font></i>: 
<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">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #)
 ;<br/>

<b>thus </b>the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> = 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g'</font>
<b>by </b><i/>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f'</font>
<b>by </b><i>, , <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i/>
;<br/>


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

<b>thus </b><a NAME="E3:12_3_1"/>
( the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> implies <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">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) )
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:12_3"/>
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #)  st the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/>( the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> &amp; the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_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 <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #), <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #);<br/>

<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font>, <font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>assume </b><a NAME="E2:12_3_2"/><i><font color="Green">E38</font></i>: 
the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
 ;<br/>

<b>then </b><i><font color="Green">E39</font></i>: the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g'</font> = 
the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i/>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f'</font>
<b>by </b><i/>
;<br/>
<a NAME="E4:12_3_2"/><i><font color="Green">E40</font></i>: 
<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">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #)
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:12_3_2"/><b>then </b><i><font color="Green">E41</font></i>: 
the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g'</font>,<font color="Maroon">f'</font></span><a href="cat_2.html#K13">]</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E6:12_3_2"/><i><font color="Green">E42</font></i>: 
the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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> is   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #)
 
<b>by </b><i>, <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>
<a NAME="E7:12_3_2"/><b>then </b><i><font color="Green">E43</font></i>: 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g'</font>,<font color="Maroon">f'</font></span><a href="cat_2.html#K13">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b>the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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> = 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">A</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">A</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g'</font>,<font color="Maroon">f'</font></span><a href="cat_2.html#K13">]</a></span></span>)</span>
<b>by </b><i>, , <a class="ref" href="nattra_1.html#T2" target="_self">Th2</a></i>
<br/>.= 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f'</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i/>
;<br/><br/>

<b>thus </b>the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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> = 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">A</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">A</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g'</font>,<font color="Maroon">f'</font></span><a href="cat_2.html#K13">]</a></span></span>)</span>
<b>by </b><i>, , <a class="ref" href="nattra_1.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g'</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font>
<b>by </b><i/>
;<br/>


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

<b>thus </b><a NAME="E3:12_3"/>
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 <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #)  st the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> &amp; the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/>the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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="12_3_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 <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #), <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #), <font color="Maroon">h</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #);<br/>

<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font>, <font color="Maroon">f'</font> = <font color="Maroon">f</font>, <font color="Maroon">h'</font> = <font color="Maroon">h</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E2:12_3_3"/><i><font color="Green">E44</font></i>: 
the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E3:12_3_3"/><i><font color="Green">E45</font></i>: 
the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
 ;<br/>

<i><font color="Green">E46</font></i>: the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">h'</font> = 
the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font>
<b>by </b><i>, </i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g'</font>
<b>by </b><i/>
;<br/>
<i><font color="Green">E47</font></i>: the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g'</font> = 
the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i>, </i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f'</font>
<b>by </b><i/>
;<br/>
<a NAME="E6:12_3_3"/><i><font color="Green">E49</font></i>: 
<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">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #)
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:12_3_3"/><b>then </b><i><font color="Green">E50</font></i>: 
the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">h'</font>,<font color="Maroon">g'</font></span><a href="cat_2.html#K13">]</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E8:12_3_3"/><i><font color="Green">E51</font></i>: 
<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">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #)
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:12_3_3"/><b>then </b><i><font color="Green">E88</font></i>: 
the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g'</font>,<font color="Maroon">f'</font></span><a href="cat_2.html#K13">]</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>reconsider </b><font color="Maroon">gf</font> = the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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>, <font color="Maroon">hg</font> = the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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> as   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <b>by </b><i><a class="ref" href="nattra_1.html#T2" target="_self">Th2</a>, , <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>
<a NAME="E11:12_3_3"/>
( <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g'</font>,<font color="Maroon">f'</font></span><a href="cat_2.html#K13">]</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">A</font> &amp; <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">h'</font>,<font color="Maroon">g'</font></span><a href="cat_2.html#K13">]</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">A</font> )
 
<b>by </b><i>, , <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">gf'</font> = the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g'</font>,<font color="Maroon">f'</font></span><a href="cat_2.html#K13">]</a></span>, <font color="Maroon">hg'</font> = the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">h'</font>,<font color="Maroon">g'</font></span><a href="cat_2.html#K13">]</a></span> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>
the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font> = 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g'</font>
<b>by </b><i>, </i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">gf'</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">gf</font>
<b>by </b><i>, </i>
;<br/>
<a NAME="E14:12_3_3"/><b>then </b><i><font color="Green">E93</font></i>: 
<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 <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #)
 
<b>by </b><i/>;<br/>
the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">hg</font> = 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">hg'</font>
<b>by </b><i>, </i>
<br/>.= 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g'</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i>, </i>
;<br/>
<a NAME="E16:12_3_3"/><b>then </b><i><font color="Green">E94</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">hg</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">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #)
 
<b>by </b><i/>;<br/>
<b>thus </b>the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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> = 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">A</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">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g'</font>,<font color="Maroon">f'</font></span><a href="cat_2.html#K13">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="txt" href="nattra_1.html#E3:12"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">A</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">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">h'</font>,<font color="Maroon">g'</font></span><a href="cat_2.html#K13">]</a></span></span>)</span>,<font color="Maroon">f'</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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="nattra_1.html#E4:12"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/>


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

<b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #);<br/>

<b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:12_3"/>
 <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #)
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:12_3"/><b>then </b><i><font color="Green">E95</font></i>: 
the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b'</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b>the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> = 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">A</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">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>
;<br/><br/>

<b>thus </b>the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> = 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">A</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">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>
;<br/>

<b>thus </b><a NAME="E9:12_3"/>
for <font color="Olive">f</font> being  <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #)  st the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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="12_3_6"><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 <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #);<br/>

<b>assume </b><a NAME="E1:12_3_6"/><i><font color="Green">E96</font></i>: 
the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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/>

<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:12_3_6"/><i><font color="Green">E97</font></i>: 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 
<b>by </b><i>, </i>;<br/>
the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> = 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f'</font>
<b>by </b><i/>
<br/>.= 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">A</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">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, </i>
;<br/>
<a NAME="E5:12_3_6"/><b>then </b>
<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">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #)
 
<b>by </b><i/>;<br/>
<b>hence </b>the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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> = 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b'</font></span>)</span>,<font color="Maroon">f'</font></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>
;<br/><br/>


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

<b>let </b><font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #);<br/>

<b>assume </b><a NAME="E10:12_3"/><i><font color="Green">E98</font></i>: 
the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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/>

<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:12_3"/><i><font color="Green">E99</font></i>: 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 
<b>by </b><i>, </i>;<br/>
the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> = 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g'</font>
<b>by </b><i/>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">A</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">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, </i>
;<br/>
<a NAME="E14:12_3"/><b>then </b>
<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">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #)
 
<b>by </b><i/>;<br/>
<b>hence </b>the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</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">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</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 <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b'</font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">g</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">C</font> =  <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) as   <a href="cat_1.html#NM4">Category</a> ;<br/>
<a NAME="E10:12"/>
<font color="Maroon">C</font> is    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:12_4"/>
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_2.html#D4">CAT_2:def 4</a><br/>

<b>thus </b><a NAME="E2:12_4"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">a'</font>, <font color="Olive">b'</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a'</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b'</font> holds <br/> <a href="cat_1.html#K6">Hom</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="tarski.html#R1">c=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">a'</font>,<font color="Olive">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>, <font color="Maroon">b'</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E1:12_4_1"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> )
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:12_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/>

<a NAME="E3:12_4_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> : (  <a href="cat_1.html#K2">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E5:12_4_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E6:12_4_1"/><i><font color="Green">E102</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:12_4_1"/><i><font color="Green">E103</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E9:12_4_1"/>
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E10:12_4_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font> : (  <a href="cat_1.html#K2">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E11:12_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>
 ;<br/>


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

<b>thus </b><a NAME="E3:12_4"/>
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T88">RELAT_1:88</a></i>;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E4:12_4"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 ;<br/>

<a NAME="E5:12_4"/><i><font color="Green">E105</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E6:12_4"/>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">a'</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:12"/>
 <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">DOM</font>,<font color="Maroon">COD</font>,<font color="Maroon">COMP</font>,<font color="Maroon">ID</font> #) is    <a href="cat_2.html#M3">Subcategory</a> of <font color="Maroon">A</font>
 ;<br/>


</div>
