<?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">B</font> be   <a href="cat_1.html#NM4">Category</a>;<br/>



<b>assume </b><a NAME="E1:64"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="isocat_1.html#NR5" target="_self">are_equivalent</a> 
 ;<br/>

<b>let </b><font color="Maroon">F</font> be    <a href="isocat_1.html#M1" target="_self">Equivalence</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>

<b>consider </b><font color="Maroon">G</font> being    <a href="isocat_1.html#M1" target="_self">Equivalence</a> of <font color="Maroon">B</font>,<font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:64"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font> <a href="nattra_1.html#NR4" target="_self">~=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:64"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">F</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G</font> <a href="nattra_1.html#NR4" target="_self">~=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:64"/><i><font color="Green">E23</font></i>: 
 <a href="cat_1.html#K15">id</a> <font color="Maroon">A</font> <a href="nattra_1.html#NR4" target="_self">~=</a> <font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="nattra_1.html#T31">NATTRA_1:31</a></i>;<br/>
<a NAME="E6:64"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="cat_1.html#K15">id</a> <font color="Maroon">A</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D11">NATTRA_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">s</font> being    <a href="nattra_1.html#M2">natural_transformation</a> of  <a href="cat_1.html#K15">id</a> <font color="Maroon">A</font>,<font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font><b> such that </b><br/><a NAME="E8:64"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">s</font> is <a href="nattra_1.html#V1">invertible</a>
 <b>by </b><i>, <a class="ref" href="nattra_1.html#D11">NATTRA_1:def 11</a></i>;<br/>
<a NAME="E9:64"/><i><font color="Green">E55</font></i>: 
 <a href="cat_1.html#K15">id</a> <font color="Maroon">B</font> <a href="nattra_1.html#NR4" target="_self">~=</a> <font color="Maroon">F</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="nattra_1.html#T31">NATTRA_1:31</a></i>;<br/>
<a NAME="E10:64"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="cat_1.html#K15">id</a> <font color="Maroon">B</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D11">NATTRA_1:def 11</a></i>;<br/>
<a NAME="E11:64"/><i><font color="Green">E59</font></i>: 
 ex <font color="Olive">t</font> being   <a href="nattra_1.html#M2">natural_transformation</a> of  <a href="cat_1.html#K15">id</a> <font color="Maroon">B</font>,<font color="Maroon">F</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G</font> st <font color="Olive">t</font> is <a href="nattra_1.html#V1">invertible</a>
 
<b>by </b><i>, <a class="ref" href="nattra_1.html#D11">NATTRA_1:def 11</a></i>;<br/>
<a NAME="E12:64"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">G</font> is <a href="cat_1.html#V10">faithful</a>
 
<b>by </b><i>, <a class="ref" href="isocat_1.html#T9" target="_self">Th9</a></i>;<br/>
<b>thus </b><a NAME="E13:64"/>
<font color="Maroon">F</font> is <a href="cat_1.html#V9">full</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_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">A</font>, <font color="Maroon">a'</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D23">CAT_1:def 23</a><br/>

<b>assume </b><a NAME="E1:64_1"/><i><font color="Green">E61</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a'</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">g</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font>,<font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a'</font>;<br/>

<a NAME="E2:64_1"/><i><font color="Green">E80</font></i>: 
( <span class="p1">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">A</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">A</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T118">CAT_1:118</a></i>;<br/>
<a NAME="E3:64_1"/><i><font color="Green">E85</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a'</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T113">CAT_1:113</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">G</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font> as    <a href="cat_1.html#M1">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a'</font> ;<br/>
<a NAME="E5:64_1"/><i><font color="Green">E86</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">A</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="isocat_1.html#T10" target="_self">Th10</a>, </i>;<br/>
<a NAME="E6:64_1"/><i><font color="Green">E87</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a'</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="isocat_1.html#D2" target="_self">Def2</a>, , <a class="ref" href="cat_1.html#T126">CAT_1:126</a></i>;<br/>
<a NAME="E7:64_1"/>
<font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="nattra_1.html#D11">NATTRA_1:def 11</a></i>;<br/>
<a NAME="E8:64_1"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">A</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a'</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E9:64_1"/><i><font color="Green">E89</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">A</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a'</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="isocat_1.html#E10:64"><i><font color="Green">E57</font></i></a>, , <a class="ref" href="cat_1.html#T52">CAT_1:52</a></i>;<br/>
<b>hence </b><a NAME="E10:64_1"/><i><font color="Green">E90</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">a'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="isocat_1.html#E9:64"><i><font color="Green">E55</font></i></a>, <a class="txt" href="isocat_1.html#E11:64"><i><font color="Green">E59</font></i></a>, <a class="ref" href="cat_1.html#T52">CAT_1:52</a></i>;<br/>

<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> <b>by </b><i><a class="ref" href="cat_1.html#T118">CAT_1:118</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f2</font> = <font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a'</font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">a'</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a'</font> <b>by </b><i><a class="ref" href="cat_1.html#T118">CAT_1:118</a></i>;<br/>
<b>take </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="cat_1.html#K11">"</a> </span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K9">*</a> <font color="Maroon">f1</font></span>)</span>;<br/>

<a NAME="E13:64_1"/><i><font color="Green">E100</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">A</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a'</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="isocat_1.html#T10" target="_self">Th10</a>, </i>;<br/>
<a NAME="E14:64_1"/>
<font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> is <a href="cat_1.html#V5">invertible</a>
 
<b>by </b><i><a class="ref" href="isocat_1.html#T11" target="_self">Th11</a>, <a class="ref" href="nattra_1.html#D10">NATTRA_1:def 10</a></i>;<br/>
<a NAME="E15:64_1"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> is <a href="cat_1.html#V4">epi</a>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E10:64"><i><font color="Green">E57</font></i></a>, <a class="ref" href="cat_1.html#T73">CAT_1:73</a></i>;<br/>
<a NAME="E16:64_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a'</font> is <a href="cat_1.html#V5">invertible</a>
 
<b>by </b><i><a class="ref" href="isocat_1.html#T11" target="_self">Th11</a>, <a class="ref" href="nattra_1.html#D10">NATTRA_1:def 10</a></i>;<br/>
<a NAME="E17:64_1"/><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">A</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a'</font></span>)</span> <a href="cat_1.html#K11">"</a> </span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E9:64"><i><font color="Green">E55</font></i></a>, <a class="txt" href="isocat_1.html#E1:64_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="nattra_1.html#T16">NATTRA_1:16</a></i>;<br/>
<font color="Maroon">h</font> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a'</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E12:64"><i><font color="Green">E60</font></i></a>, <a class="ref" href="cat_1.html#T57">CAT_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a'</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a'</font></span>)</span> <a href="cat_1.html#K11">"</a> </span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="isocat_1.html#D3" target="_self">Def3</a>, <a class="ref" href="isocat_1.html#T13" target="_self">Th13</a>, <a class="ref" href="cat_1.html#D14">CAT_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a'</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">A</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E11:64"><i><font color="Green">E59</font></i></a>, <a class="txt" href="isocat_1.html#E12:64"><i><font color="Green">E60</font></i></a>, <a class="ref" href="isocat_1.html#D3" target="_self">Def3</a>, <a class="ref" href="isocat_1.html#T16" target="_self">Th16</a>, <a class="ref" href="cat_1.html#T54">CAT_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="isocat_1.html#T10" target="_self">Th10</a>, <a class="txt" href="isocat_1.html#E1:64_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="nattra_1.html#D8">NATTRA_1:def 8</a></i>
;<br/>
<a NAME="E19:64_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E10:64"><i><font color="Green">E57</font></i></a>, , , <a class="ref" href="cat_1.html#T65">CAT_1:65</a></i>;<br/>
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> = 
<font color="Maroon">G</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="isocat_1.html#D2" target="_self">Def2</a>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i><a class="txt" href="isocat_1.html#E1:64_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="isocat_1.html#T17" target="_self">Th17</a>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E1:64_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
;<br/>
<b>hence </b><font color="Maroon">g</font> = 
<font color="Maroon">F</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="isocat_1.html#D2" target="_self">Def2</a>, <a class="ref" href="cat_1.html#D24">CAT_1:def 24</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i><a class="txt" href="isocat_1.html#E1:64_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
;<br/><br/>


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

<b>thus </b><a NAME="E14:64"/>
<font color="Maroon">F</font> is <a href="cat_1.html#V10">faithful</a>
 <b>by </b><i>, <a class="ref" href="isocat_1.html#T9" target="_self">Th9</a></i>;<br/>

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

<b>take </b>
<font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font>
;<br/>

<a NAME="E15:64"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">B</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T113">CAT_1:113</a>, <a class="ref" href="cat_1.html#T118">CAT_1:118</a></i>;<br/>
<a NAME="E16:64"/>
 <a href="cat_1.html#K15">id</a> <font color="Maroon">B</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">F</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="isocat_1.html#T12" target="_self">Th12</a>, <a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E17:64"/>
<font color="Maroon">b</font>,<font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#R1">are_isomorphic</a> 
 <b>by </b><i>, <a class="ref" href="isocat_1.html#D2" target="_self">Def2</a>, </i>;<br/>


</div>
