<?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>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/>





<a NAME="E1:39"/><i><font color="Green">E16</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K7" target="_self">export</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E2:39"/>
 <a href="isocat_2.html#K7" target="_self">export</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font> is <a href="funct_1.html#V2">one-to-one</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="isocat_1.html#D3">ISOCAT_1:def 3</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K7" target="_self">export</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">x1</font> as   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>, <font color="Maroon">t</font> being    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E4:39_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 <b>and </b><br/><a NAME="E5:39_1"/><i><font color="Green">E32</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E6:39_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K7" target="_self">export</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f2</font> = <font color="Maroon">x2</font> as   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">G1</font> being    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>, <font color="Maroon">G2</font> being    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>, <font color="Maroon">u</font> being    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">G1</font>,<font color="Maroon">G2</font><b> such that </b><br/><a NAME="E9:39_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">G1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G2</font>
 <b>and </b><br/><a NAME="E10:39_1"/><i><font color="Green">E50</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> &amp; <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E11:39_1"/>
<span class="p1">(<span class="default"><a href="isocat_2.html#K7" target="_self">export</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K7" target="_self">export</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/>

<b>then </b><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" target="_self">export</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><a href="isocat_2.html#K7" target="_self">export</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="isocat_2.html#T7" target="_self">Th7</a>, , </i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">G1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">G2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" target="_self">export</a> <font color="Maroon">u</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, </i>
;<br/>
<a NAME="E13:39_1"/><b>then </b><i><font color="Green">E51</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">G1</font></span>)</span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">G2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="isocat_2.html#K6" target="_self">export</a> <font color="Maroon">u</font> <a href="funct_2.html#R2">=</a>  <a href="isocat_2.html#K6" target="_self">export</a> <font color="Maroon">t</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E14:39_1"/><b>then </b>
(  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a>  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">G1</font> &amp;  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font> <a href="funct_2.html#R2">=</a>  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">G2</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E15:39_1"/><b>then </b>
( <font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">F2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">G2</font> )
 
<b>by </b><i><a class="ref" href="isocat_2.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E16:39_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, <a class="ref" href="isocat_2.html#T7" target="_self">Th7</a>, , , </i>;<br/>


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

<b>thus </b><a NAME="E3:39"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K7" target="_self">export</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

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

<b>assume </b><a NAME="E4:39"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">m</font> as   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>, <a href="nattra_1.html#K12">Functors</a> <font color="Maroon">B</font>,<font color="Maroon">C</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>, <a href="nattra_1.html#K12">Functors</a> <font color="Maroon">B</font>,<font color="Maroon">C</font>, <font color="Maroon">t</font> being    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E7:39"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 <b>and </b><br/><a NAME="E8:39"/><i><font color="Green">E55</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">G1</font> being    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font><b> such that </b><br/><a NAME="E10:39"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a>  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">G1</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">G2</font> being    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font><b> such that </b><br/><a NAME="E12:39"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">F2</font> <a href="funct_2.html#R2">=</a>  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">G2</font>
 <b>by </b><i/>;<br/>
<a NAME="E13:39"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">G1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G2</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>consider </b><font color="Maroon">u</font> being    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">G1</font>,<font color="Maroon">G2</font><b> such that </b><br/><a NAME="E15:39"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">t</font> <a href="funct_2.html#R2">=</a>  <a href="isocat_2.html#K6" target="_self">export</a> <font color="Maroon">u</font>
 <b>by </b><i>, , , </i>;<br/>
<a NAME="E16:39"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="isocat_2.html#K7" target="_self">export</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="isocat_2.html#T7" target="_self">Th7</a>, , , , <a class="ref" href="isocat_2.html#T8" target="_self">Th8</a>, </i>;<br/>
<a NAME="E17:39"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11">NatTrans</a> <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="nattra_1.html#T35">NATTRA_1:35</a></i>;<br/>
<a NAME="E18:39"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K7" target="_self">export</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a>, <a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/>
<b>hence </b><a NAME="E19:39"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K7" target="_self">export</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="isocat_2.html#T9" target="_self">Th9</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


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