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

<b>let </b><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">D</font> be  <a href="cat_5.html#V3" title="CAT_5:attr.3">Categorial</a> <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/>

<b>assume </b><a NAME="E1:34"/><i><font color="Green" title="E21">A1</font></i>: 
the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c2">D</font>
 ;<br/>

<b>thus </b><a NAME="E2:34"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c2">D</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="cat_2.html#D4" title="CAT_2:def.4">CAT_2:def 4</a><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="c3">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_1"/>
<font color="Maroon" title="c3">x</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>then reconsider </b><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c3">x</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="c5">i</font> =  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">D</font> <b>by </b><i><a class="txt" href="cat_5.html#E1:34"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:34_1"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a>  &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a>  )
 
<b>by </b><i><a class="ref" href="cat_5.html#T13" target="_self" title="CAT_5:th.13">Th13</a>, <a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>;<br/>
<b>hence </b><a NAME="E5:34_1"/>
<font color="Maroon" title="c3">x</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="c2">D</font>
 ;<br/>


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

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/><b>let </b><font color="Maroon" title="c5">a'</font>, <font color="Maroon" title="c6">b'</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font>;<br/><b>assume </b><a NAME="E1:34_2"/><i><font color="Green" title="E22">A2</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a'</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b'</font> )
 ;<br/><b>thus </b><a NAME="E2:34_2"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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_2_1"/>
<font color="Maroon" title="c7">x</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> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 ;<br/>

<a NAME="E2:34_2_1"/><b>then </b>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> : (  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c8">f</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E4:34_2_1"/><i><font color="Green" title="E23">A3</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">A</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c10">B</font> = <font color="Maroon" title="c4">b</font> as   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> <b>by </b><i><a class="ref" href="cat_5.html#T12" target="_self" title="CAT_5:th.12">Th12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">F</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c10">B</font><b> such that </b><br/><a NAME="E7:34_2_1"/><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c10">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="cat_5.html#E4:34_2_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="cat_5.html#D6" target="_self" title="CAT_5:def.6">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">f</font> = <font color="Maroon" title="c8">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">D</font> <b>by </b><i><a class="txt" href="cat_5.html#E1:34"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E9:34_2_1"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a>  &amp; <font color="Maroon" title="c12">f</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c12">f</font> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B</font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E7:34_2_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="cat_5.html#T13" target="_self" title="CAT_5:th.13">Th13</a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/>
<b>hence </b><a NAME="E10:34_2_1"/>
<font color="Maroon" title="c7">x</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> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font>
 <b>by </b><i><a class="txt" href="cat_5.html#E1:34_2"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="cat_5.html#E4:34_2_1"><i><font color="Green" title="E23">A3</font></i></a></i>;<br/>


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

<a NAME="E4:34"/><i><font color="Green" title="E22">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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_3"/><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  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="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">g'</font> = <font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c7">f'</font> = <font color="Maroon" title="c5">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">D</font> <b>by </b><i><a class="txt" href="cat_5.html#E1:34"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:34_3"/><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c5">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
 
<b>by </b><i><a class="txt" href="cat_5.html#E1:34_3"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<a NAME="E5:34_3"/><b>then </b>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c4">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="c5">f</font> &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a>  &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a>  )
 
<b>by </b><i><a class="txt" href="cat_5.html#E1:34_3"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="cat_5.html#T13" target="_self" title="CAT_5:th.13">Th13</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="E6:34_3"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font>
 <b>by </b><i><a class="txt" href="cat_5.html#E4:34_3"><i><font color="Green" title="E24">A7</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>
<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="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:34_4"/><i><font color="Green" title="E23">A8</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c3">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  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="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">g'</font> = <font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c7">f'</font> = <font color="Maroon" title="c5">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">D</font> <b>by </b><i><a class="txt" href="cat_5.html#E1:34"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E4:34_4"/><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c5">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
 <b>by </b><i><a class="txt" href="cat_5.html#E1:34_4"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/><a NAME="E5:34_4"/><i><font color="Green" title="E25">A10</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a>  )
 <b>by </b><i><a class="ref" href="cat_5.html#T13" target="_self" title="CAT_5:th.13">Th13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">G</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c4">g</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a> ,<font color="Maroon" title="c4">g</font> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a> <b> such that </b><br/><a NAME="E7:34_4"/><i><font color="Green" title="E26">A11</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c8">G</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#D6" target="_self" title="CAT_5:def.6">Def6</a></i>;<br/><a NAME="E8:34_4"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">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="c4">g</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E1:34_4"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="cat_5.html#E4:34_4"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="cat_5.html#T13" target="_self" title="CAT_5:th.13">Th13</a>, <a class="ref" href="cat_1.html#T40" title="CAT_1:th.40">CAT_1:40</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">F</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c5">f</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a> ,<font color="Maroon" title="c4">g</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a> <b> such that </b><br/><a NAME="E10:34_4"/><i><font color="Green" title="E27">A12</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="cat_5.html#E5:34_4"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="cat_5.html#D6" target="_self" title="CAT_5:def.6">Def6</a></i>;<br/><b>thus </b>the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</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="c3">x</font> = 
<font color="Maroon" title="c4">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c5">f</font>
<b>by </b><i><a class="txt" href="cat_5.html#E1:34_4"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="cat_5.html#E4:34_4"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="cat_1.html#D4" title="CAT_1:def.4">CAT_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c9">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E7:34_4"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="cat_5.html#E10:34_4"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="cat_5.html#D6" target="_self" title="CAT_5:def.6">Def6</a></i>
<br/>.= 
<font color="Maroon" title="c6">g'</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c7">f'</font>
<b>by </b><i><a class="txt" href="cat_5.html#E7:34_4"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="cat_5.html#E10:34_4"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="cat_5.html#D6" target="_self" title="CAT_5:def.6">Def6</a></i>
<br/>.= 
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="txt" href="cat_5.html#E4:34"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="cat_5.html#E1:34_4"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="cat_5.html#E4:34_4"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="cat_1.html#D4" title="CAT_1:def.4">CAT_1:def 4</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:34"/>
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</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="c2">D</font>
 <b>by </b><i><a class="txt" href="cat_5.html#E4:34"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/>

<b>let </b><font color="Maroon" title="c3">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/><b>let </b><font color="Maroon" title="c4">a'</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font>;<br/>



<b>assume </b><a NAME="E7:34"/><i><font color="Green" title="E23">A13</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a'</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c3">a</font> as   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> <b>by </b><i><a class="ref" href="cat_5.html#T12" target="_self" title="CAT_5:th.12">Th12</a></i>;<br/>
<b>thus </b> <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">a</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c5">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c5">A</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="cat_5.html#D6" target="_self" title="CAT_5:def.6">Def6</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a'</font>
<b>by </b><i><a class="txt" href="cat_5.html#E7:34"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="cat_5.html#D6" target="_self" title="CAT_5:def.6">Def6</a></i>
;<br/>


</div>
