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

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c2">O</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">d</font>, <font color="Maroon" title="c5">c</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">O</font>;<br/><b>let </b><font color="Maroon" title="c6">p</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c3">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c3">M</font>;<br/><b>let </b><font color="Maroon" title="c7">i</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>;<br/>











<b>set </b><font color="Maroon" title="c8">H</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c9">B</font> =  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #);<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:34"/><i><font color="Green" title="E19">A1</font></i>: 
 <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) is    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green" title="E20">A2</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #)<br/> for <font color="Olive" title="b3">a'</font>, <font color="Olive" title="b4">b'</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a'</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b'</font> holds <br/> <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b3">a'</font>,<font color="Olive" title="b4">b'</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_2.html#D6" target="_self" title="CAT_2:def.6">CAT_2:def 6</a><br/>

<a NAME="E3:34"/><i><font color="Green" title="E21">A3</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) holds <br/> ( <font color="Maroon" title="c4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> &amp; <font color="Maroon" title="c5">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c11">f'</font> = <font color="Maroon" title="c10">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="cat_2.html#E1:34"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="cat_2.html#T14" target="_self" title="CAT_2:th.14">Th14</a></i>;<br/>
<a NAME="E2:34_1"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c11">f'</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c11">f'</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c11">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">f'</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c11">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">f'</font> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E1:34"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="cat_2.html#T15" target="_self" title="CAT_2:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E3:34_1"/>
( <font color="Maroon" title="c4">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Maroon" title="c5">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</font> )
 ;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:34_2"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span>}</span>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c10">x</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">f'</font> = <font color="Maroon" title="c11">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="cat_2.html#E1:34"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="cat_2.html#T14" target="_self" title="CAT_2:th.14">Th14</a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">a'</font> =  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c12">f'</font>;<br/>
<b>set </b><font color="Maroon" title="c14">b'</font> =  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c12">f'</font>;<br/>
<a NAME="E4:34_2_1"/>
( the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f'</font> &amp; the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f'</font> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E3:34"><i><font color="Green" title="E21">A3</font></i></a></i>;<br/>
<a NAME="E5:34_2_1"/><b>then </b>
( <font color="Maroon" title="c11">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c12">f'</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c12">f'</font></span>)</span> &amp;  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c12">f'</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c12">f'</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span>}</span>  )
 
;<br/>
<b>hence </b><a NAME="E6:34_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:34_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c11">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:34_2"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 <b>and </b><br/><a NAME="E5:34_2"/><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">a'</font>, <font color="Maroon" title="c13">b'</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E7:34_2"/><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c12">a'</font>,<font color="Maroon" title="c13">b'</font>
 <b>and </b><br/><a NAME="E8:34_2"/><i><font color="Green" title="E25">A7</font></i>: 
( <font color="Maroon" title="c12">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Maroon" title="c13">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> )
 <b>by </b><i><a class="txt" href="cat_2.html#E5:34_2"><i><font color="Green" title="E23">A5</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">a</font> = <font color="Maroon" title="c12">a'</font>, <font color="Maroon" title="c15">b</font> = <font color="Maroon" title="c13">b'</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) <b>by </b><i><a class="txt" href="cat_2.html#E8:34_2"><i><font color="Green" title="E25">A7</font></i></a></i>;<br/><a NAME="E10:34_2"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c12">a'</font>,<font color="Maroon" title="c13">b'</font>
 <b>by </b><i><a class="txt" href="cat_2.html#E2:34"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E11:34_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="cat_2.html#E4:34_2"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="cat_2.html#E7:34_2"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:34"/><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">d'</font> = the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font>, <font color="Maroon" title="c11">c'</font> = the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">O</font> <b>by </b><i><a class="ref" href="cat_2.html#T29" target="_self" title="CAT_2:th.29">Th29</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">f</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:34_3_1"/>
<font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">f</font>
 <b>by </b><i><a class="txt" href="cat_2.html#E3:34"><i><font color="Green" title="E21">A3</font></i></a></i>;<br/><b>thus </b><a NAME="E2:34_3_1"/>
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 ;<br/><a NAME="E3:34_3_1"/>
<font color="Maroon" title="c12">f</font> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="cat_2.html#E1:34"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="cat_2.html#T14" target="_self" title="CAT_2:th.14">Th14</a></i>;<br/><a NAME="E4:34_3_1"/><b>then </b>
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>
 ;<br/><b>hence </b><a NAME="E5:34_3_1"/>
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:34_3"/>
<font color="Maroon" title="c4">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">d'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:34"/>
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">f</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:34_4_1"/>
<font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">f</font>
 <b>by </b><i><a class="txt" href="cat_2.html#E3:34"><i><font color="Green" title="E21">A3</font></i></a></i>;<br/><b>thus </b><a NAME="E2:34_4_1"/>
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 ;<br/><a NAME="E3:34_4_1"/>
<font color="Maroon" title="c12">f</font> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="cat_2.html#E1:34"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="cat_2.html#T14" target="_self" title="CAT_2:th.14">Th14</a></i>;<br/><a NAME="E4:34_4_1"/><b>then </b>
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>
 ;<br/><b>hence </b><a NAME="E5:34_4_1"/>
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:34_4"/>
<font color="Maroon" title="c5">c</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:34"/>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>

<b>set </b><font color="Maroon" title="c12">p'</font> = the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_5"><b>now </b></a><div class="add"><a NAME="E1:34_5"/><i><font color="Green" title="E23">A9</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c3">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><a NAME="E2:34_5"/><i><font color="Green" title="E24">A10</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c3">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E3:34_5"/><i><font color="Green" title="E25">A11</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_5_1"/><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font></span>)</span>
 ;<br/>

<a NAME="E2:34_5_1"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c3">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="cat_2.html#E2:34_5"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">g</font>, <font color="Maroon" title="c15">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font><b> such that </b><br/><a NAME="E4:34_5_1"/><i><font color="Green" title="E27">A13</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">f</font> = <font color="Maroon" title="c15">f</font>, <font color="Maroon" title="c17">g</font> = <font color="Maroon" title="c14">g</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) ;<br/>
<b>reconsider </b><font color="Maroon" title="c18">f'</font> = <font color="Maroon" title="c16">f</font>, <font color="Maroon" title="c19">g'</font> = <font color="Maroon" title="c17">g</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="cat_2.html#E1:34"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="cat_2.html#T14" target="_self" title="CAT_2:th.14">Th14</a></i>;<br/>
<a NAME="E7:34_5_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">g</font>,<font color="Maroon" title="c16">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="cat_2.html#E2:34_5"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="cat_2.html#E1:34_5_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="cat_2.html#E4:34_5_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:34_5_1"/><b>then </b>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c16">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c18">f'</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c19">g'</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c19">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c18">f'</font> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E1:34"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="cat_2.html#T15" target="_self" title="CAT_2:th.15">Th15</a>, <a class="ref" href="cat_1.html#T40" title="CAT_1:th.40">CAT_1:40</a></i>;<br/>
<b>hence </b><a NAME="E9:34_5_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="cat_2.html#E1:34"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="cat_2.html#E4:34_5_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="cat_1.html#T40" title="CAT_1:th.40">CAT_1:40</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:34_5"/>
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) <a href="funct_4.html#NR1" title="FUNCT_4:NR.1">&lt;=</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="cat_2.html#E1:34"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="cat_2.html#D4" target="_self" title="CAT_2:def.4">Def4</a></i>;<br/><a NAME="E5:34_5"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/><a NAME="E6:34_5"/><b>then </b><i><font color="Green" title="E26">A14</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font></span>)</span>
 <b>by </b><i><a class="txt" href="cat_2.html#E1:34_5"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="cat_2.html#E2:34_5"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/><b>hence </b><a NAME="E7:34_5"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font></span>)</span>
 <b>by </b><i><a class="txt" href="cat_2.html#E3:34_5"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E8:34_5"/><i><font color="Green" title="E27">A15</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">p</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c14">g</font>, <font color="Maroon" title="c15">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font><b> such that </b><br/><a NAME="E10:34_5"/><i><font color="Green" title="E28">A16</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="cat_2.html#E1:34_5"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">f</font> = <font color="Maroon" title="c15">f</font>, <font color="Maroon" title="c17">g</font> = <font color="Maroon" title="c14">g</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) ;<br/><b>reconsider </b><font color="Maroon" title="c18">f'</font> = <font color="Maroon" title="c16">f</font>, <font color="Maroon" title="c19">g'</font> = <font color="Maroon" title="c17">g</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="cat_2.html#E1:34"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="cat_2.html#T14" target="_self" title="CAT_2:th.14">Th14</a></i>;<br/><a NAME="E13:34_5"/><i><font color="Green" title="E29">A17</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c16">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c18">f'</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c19">g'</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c16">f</font> )
 <b>by </b><i><a class="txt" href="cat_2.html#E1:34"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="cat_2.html#E8:34_5"><i><font color="Green" title="E27">A15</font></i></a>, <a class="txt" href="cat_2.html#E10:34_5"><i><font color="Green" title="E28">A16</font></i></a>, <a class="ref" href="cat_2.html#T15" target="_self" title="CAT_2:th.15">Th15</a>, <a class="ref" href="cat_1.html#T40" title="CAT_1:th.40">CAT_1:40</a></i>;<br/><a NAME="E14:34_5"/>
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c17">g</font>,<font color="Maroon" title="c16">f</font>
 <b>by </b><i><a class="txt" href="cat_2.html#E10:34_5"><i><font color="Green" title="E28">A16</font></i></a></i>;<br/><b>hence </b><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> = 
<font color="Maroon" title="c17">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c16">f</font>
<b>by </b><i><a class="txt" href="cat_2.html#E1:34"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="cat_1.html#T41" title="CAT_1:th.41">CAT_1:41</a>, <a class="txt" href="cat_2.html#E13:34_5"><i><font color="Green" title="E29">A17</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c19">g'</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c18">f'</font>
<b>by </b><i><a class="txt" href="cat_2.html#E1:34"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="cat_2.html#E13:34_5"><i><font color="Green" title="E29">A17</font></i></a>, <a class="ref" href="cat_2.html#T17" target="_self" title="CAT_2:th.17">Th17</a></i>
<br/>.= 
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">g'</font>,<font color="Maroon" title="c18">f'</font>
<b>by </b><i><a class="txt" href="cat_2.html#E13:34_5"><i><font color="Green" title="E29">A17</font></i></a>, <a class="ref" href="cat_1.html#T41" title="CAT_1:th.41">CAT_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="txt" href="cat_2.html#E6:34_5"><i><font color="Green" title="E26">A14</font></i></a>, <a class="txt" href="cat_2.html#E8:34_5"><i><font color="Green" title="E27">A15</font></i></a>, <a class="txt" href="cat_2.html#E10:34_5"><i><font color="Green" title="E28">A16</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:34"/>
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c13">i'</font> = the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">O</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font> <b>by </b><i><a class="txt" href="cat_2.html#E5:34"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="cat_2.html#T29" target="_self" title="CAT_2:th.29">Th29</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">a</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">O</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_6_1"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon" title="c15">a'</font> = <font color="Maroon" title="c14">a</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ;<br/><b>reconsider </b><font color="Maroon" title="c16">b</font> = <font color="Maroon" title="c14">a</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) ;<br/><a NAME="E3:34_6_1"/>
(  <a href="cat_1.html#K5" title="CAT_1:func.5">id</a> <font color="Maroon" title="c16">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c15">a'</font> &amp;  <a href="cat_1.html#K5" title="CAT_1:func.5">id</a> <font color="Maroon" title="c16">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">b</font> &amp;  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c15">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">a'</font> )
 <b>by </b><i><a class="txt" href="cat_2.html#E1:34"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="cat_2.html#D4" target="_self" title="CAT_2:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E4:34_6_1"/>
<font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font>
 ;<br/><b>thus </b><a NAME="E5:34_6_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">O</font>
 ;<br/><a NAME="E6:34_6_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>
 ;<br/><b>hence </b><a NAME="E7:34_6_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:34_6"/>
<font color="Maroon" title="c7">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:34"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">O</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


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