<?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/><b>let </b><font color="Maroon">F1</font> be    <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> be    <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>;<br/>







<b>assume </b><a NAME="E1:37"/><i><font color="Green">E16</font></i>: 
 <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font>
 ;<br/>

<b>let </b><font color="Maroon">t</font> be    <a href="nattra_1.html#M2">natural_transformation</a> of  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font>, <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">a</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> holds <br/><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Olive">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Olive">a</font></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<font color="Maroon">a<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>;<br/>
<div><i><font color="Green">E17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>
 ;<br/><b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">o</font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font> ;<br/><b>consider </b><font color="Maroon">f1</font> being    <a href="cat_1.html#M2">Functor</a> of <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">B</font>,<font color="Maroon">C</font>, <font color="Maroon">tau</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="E4:37_1"/>
<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="E5:37_1"/><i><font color="Green">E32</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a'</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">tau</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">tau</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">m</font>;<br/><b>thus </b><a NAME="E7:37_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">B</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/><b>thus </b><a NAME="E8:37_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">o</font>,<font color="Maroon">m</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_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>;<br/>

<b>assume </b><a NAME="E1:37_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:37_1_1"/>
 <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font> <a href="nattra_1.html#R1">is_transformable_to</a>  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font>
 
<b>by </b><i>, <a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/>
<a NAME="E3:37_1_1"/><b>then </b>
( <font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> is    <a href="cat_1.html#M1">Morphism</a> of <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<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> <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="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</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="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/>
<a NAME="E4:37_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> &amp; <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> )
 
<b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a>, , <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<b>hence </b><a NAME="E5:37_1_1"/>
<font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon">t'</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>, <a href="fraenkel.html#K1">Funcs</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">B</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E4:37"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">o</font>,<font color="Maroon">t'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<a NAME="E5:37"/><i><font color="Green">E51</font></i>: 
the <a href="cat_1.html#U1">Objects</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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="cat_2.html#T33">CAT_2:33</a></i>;<br/>
<a NAME="E6:37"/>
the <a href="cat_1.html#U1">Objects</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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="cat_2.html#T33">CAT_2:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">u'</font> =  <a href="isocat_2.html#K1" target="_self">uncurry</a> <font color="Maroon">t'</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U1">Objects</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>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> ;<br/>
<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">ab</font> be   <a href="cat_1.html#NM1">Object</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>;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E2:37_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">ab</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="cat_2.html#K12">]</a></span>
 <b>by </b><i>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><a NAME="E3:37_2"/>
( <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/><a NAME="E4:37_2"/><b>then </b>
<font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</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"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t'</font> <a href="cat_2.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E5:37_2"/><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"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t'</font> <a href="cat_2.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <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">B</font>,<font color="Maroon">C</font></span>)</span>
 ;<br/><a NAME="E6:37_2"/><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"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t'</font> <a href="cat_2.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11">NatTrans</a> <font color="Maroon">B</font>,<font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/><b>then consider </b><font color="Maroon">G1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>, <font color="Maroon">G2</font> being    <a href="cat_1.html#M2">Functor</a> of <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">G1</font>,<font color="Maroon">G2</font><b> such that </b><br/><a NAME="E8:37_2"/><i><font color="Green">E57</font></i>: 
<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"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t'</font> <a href="cat_2.html#K1">.</a> <font color="Maroon">a</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"><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">t''</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E9:37_2"/><i><font color="Green">E58</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><a class="ref" href="nattra_1.html#D15">NATTRA_1:def 15</a></i>;<br/><a NAME="E10:37_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">G1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">G2</font>
 <b>by </b><i>, <a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/><a NAME="E11:37_2"/><i><font color="Green">E60</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</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"><font color="Maroon">G1</font>,<font color="Maroon">G2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">t'</font> <a href="cat_2.html#K1">.</a> <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">t''</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><i><font color="Green">E61</font></i>: <font color="Maroon">u'</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon">t'</font> <a href="cat_2.html#K1">.</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">t''</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>
<br/>.= 
<font color="Maroon">t''</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="isocat_2.html#T8" target="_self">Th8</a>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
;<br/><i><font color="Green">E62</font></i>: <font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="cat_2.html#K12">]</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">G1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="isocat_2.html#T9" target="_self">Th9</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>
;<br/><i><font color="Green">E65</font></i>: <font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="cat_2.html#K12">]</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">G2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="isocat_2.html#T9" target="_self">Th9</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>
;<br/><a NAME="E15:37_2"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</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#T30">ISOCAT_1:30</a></i>;<br/><b>hence </b><a NAME="E16:37_2"/>
<font color="Maroon">u'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ab</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">ab</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">ab</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="isocat_2.html#D1" target="_self">Def1</a>, , <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:37"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">F1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">F2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><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 <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>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a><br/>

<b>thus </b><a NAME="E1:37_3"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</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> 
 <b>by </b><i><a class="ref" href="isocat_2.html#T7" target="_self">Th7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:37"/>
<font color="Maroon">u'</font> is    <a href="nattra_1.html#M1">transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:37_4"/>
<font color="Maroon">F1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">F2</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D3">NATTRA_1:def 3</a><br/>

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

<a NAME="E2:37_4"/>
<font color="Maroon">u'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="isocat_2.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E3:37_4"/>
<font color="Maroon">u'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> is    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font>,<font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u'</font> as    <a href="nattra_1.html#M1">transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font> ;<br/>
<div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">ab1</font> be   <a href="cat_1.html#NM1">Object</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">ab2</font> be   <a href="cat_1.html#NM1">Object</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>;<br/><b>assume </b><a NAME="E1:37_5"/><i><font color="Green">E68</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">ab1</font>,<font color="Maroon">ab2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">a1</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>, <font color="Maroon">b1</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E3:37_5"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">ab1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span>
 <b>by </b><i>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><a NAME="E4:37_5"/>
( <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a1</font> &amp; <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a1</font> )
 <b>by </b><i/>;<br/><a NAME="E5:37_5"/><b>then </b>
<font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a1</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"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a1</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a1</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t'</font> <a href="cat_2.html#K1">.</a> <font color="Maroon">a1</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E6:37_5"/><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"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a1</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a1</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t'</font> <a href="cat_2.html#K1">.</a> <font color="Maroon">a1</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <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">B</font>,<font color="Maroon">C</font></span>)</span>
 ;<br/><a NAME="E7:37_5"/><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"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a1</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a1</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t'</font> <a href="cat_2.html#K1">.</a> <font color="Maroon">a1</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11">NatTrans</a> <font color="Maroon">B</font>,<font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/><b>then consider </b><font color="Maroon">G1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>, <font color="Maroon">H1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>, <font color="Maroon">t1</font> being    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">G1</font>,<font color="Maroon">H1</font><b> such that </b><br/><a NAME="E9:37_5"/><i><font color="Green">E70</font></i>: 
<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"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a1</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a1</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t'</font> <a href="cat_2.html#K1">.</a> <font color="Maroon">a1</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"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">H1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E10:37_5"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">G1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">H1</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D15">NATTRA_1:def 15</a></i>;<br/><a NAME="E11:37_5"/><i><font color="Green">E74</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a1</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"><font color="Maroon">G1</font>,<font color="Maroon">H1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">t'</font> <a href="cat_2.html#K1">.</a> <font color="Maroon">a1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">t1</font> )
 <b>by </b><i><a class="ref" href="isocat_2.html#T9" target="_self">Th9</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E12:37_5"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>consider </b><font color="Maroon">a2</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>, <font color="Maroon">b2</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E14:37_5"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">ab2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span>
 <b>by </b><i>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><a NAME="E15:37_5"/>
( <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a2</font> &amp; <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a2</font> )
 <b>by </b><i/>;<br/><a NAME="E16:37_5"/><b>then </b>
<font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a2</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"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a2</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t'</font> <a href="cat_2.html#K1">.</a> <font color="Maroon">a2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E17:37_5"/><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"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a2</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t'</font> <a href="cat_2.html#K1">.</a> <font color="Maroon">a2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <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">B</font>,<font color="Maroon">C</font></span>)</span>
 ;<br/><a NAME="E18:37_5"/><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"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a2</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t'</font> <a href="cat_2.html#K1">.</a> <font color="Maroon">a2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11">NatTrans</a> <font color="Maroon">B</font>,<font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/><b>then consider </b><font color="Maroon">G2</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>, <font color="Maroon">H2</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>, <font color="Maroon">t2</font> being    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">G2</font>,<font color="Maroon">H2</font><b> such that </b><br/><a NAME="E20:37_5"/><i><font color="Green">E77</font></i>: 
<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"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a2</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t'</font> <a href="cat_2.html#K1">.</a> <font color="Maroon">a2</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"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">G2</font>,<font color="Maroon">H2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t2</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E21:37_5"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">G2</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D15">NATTRA_1:def 15</a></i>;<br/><a NAME="E22:37_5"/><i><font color="Green">E81</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a2</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"><font color="Maroon">G2</font>,<font color="Maroon">H2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">t'</font> <a href="cat_2.html#K1">.</a> <font color="Maroon">a2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">t2</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E23:37_5"/><b>then </b><i><font color="Green">E82</font></i>: 
( <font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">G2</font> &amp; <font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">ab1</font>,<font color="Maroon">ab2</font>;<br/><a NAME="E24:37_5"/>
( <font color="Maroon">f</font> is   <a href="cat_1.html#NM2">Morphism</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> &amp; the <a href="cat_1.html#U2">Morphisms</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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="cat_2.html#T33">CAT_2:33</a></i>;<br/><b>then consider </b><font color="Maroon">g</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font>, <font color="Maroon">h</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E26:37_5"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="cat_2.html#K13">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><a NAME="E27:37_5"/><i><font color="Green">E92</font></i>: 
(  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font> <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_2.html#T8" target="_self">Th8</a>, , </i>;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">a1</font>,<font color="Maroon">a2</font> <b>by </b><i>, <a class="ref" href="isocat_2.html#T8" target="_self">Th8</a>, , , </i>;<br/><b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">b1</font>,<font color="Maroon">b2</font> <b>by </b><i>, <a class="ref" href="isocat_2.html#T8" target="_self">Th8</a>, , , </i>;<br/><a NAME="E30:37_5"/><i><font color="Green">E93</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/><a NAME="E31:37_5"/><b>then </b><i><font color="Green">E94</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>, <a class="txt" href="isocat_2.html#E5:37"><i><font color="Green">E51</font></i></a>, <a class="ref" href="isocat_2.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E32:37_5"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">G1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">G2</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/><a NAME="E33:37_5"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">H1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="txt" href="isocat_2.html#E5:37"><i><font color="Green">E51</font></i></a>, <a class="txt" href="isocat_2.html#E8:37"><i><font color="Green">E54</font></i></a>, <a class="ref" href="isocat_2.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E34:37_5"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">H1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/><a NAME="E35:37_5"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">G1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">H1</font>
 <b>by </b><i>, <a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/><a NAME="E36:37_5"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">G2</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">F1</font> <a href="isocat_2.html#K4" target="_self">?-</a> <font color="Maroon">g</font> as    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">G1</font>,<font color="Maroon">G2</font> <b>by </b><i><a class="ref" href="isocat_2.html#D1" target="_self">Def1</a>, <a class="txt" href="isocat_2.html#E5:37"><i><font color="Green">E51</font></i></a>, <a class="txt" href="isocat_2.html#E8:37"><i><font color="Green">E54</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">F2</font> <a href="isocat_2.html#K4" target="_self">?-</a> <font color="Maroon">g</font> as    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">H1</font>,<font color="Maroon">H2</font> <b>by </b><i><a class="ref" href="isocat_2.html#D1" target="_self">Def1</a>, <a class="txt" href="isocat_2.html#E5:37"><i><font color="Green">E51</font></i></a>, <a class="txt" href="isocat_2.html#E8:37"><i><font color="Green">E54</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E39:37_5"/><i><font color="Green">E100</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</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#T30">ISOCAT_1:30</a></i>;<br/><a NAME="E40:37_5"/><i><font color="Green">E101</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</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#T30">ISOCAT_1:30</a></i>;<br/><a NAME="E41:37_5"/><i><font color="Green">E138</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</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="cat_1.html#T126">CAT_1:126</a></i>;<br/><a NAME="E42:37_5"/><i><font color="Green">E139</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</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#T30">ISOCAT_1:30</a></i>;<br/><a NAME="E43:37_5"/><i><font color="Green">E140</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</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#T30">ISOCAT_1:30</a></i>;<br/><a NAME="E44:37_5"/><i><font color="Green">E141</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</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="cat_1.html#T126">CAT_1:126</a></i>;<br/><a NAME="E45:37_5"/><i><font color="Green">E142</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b1</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#T30">ISOCAT_1:30</a></i>;<br/><a NAME="E46:37_5"/><i><font color="Green">E143</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">ab1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">ab1</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="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/><a NAME="E47:37_5"/><i><font color="Green">E146</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">ab1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">ab2</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="cat_1.html#T126">CAT_1:126</a></i>;<br/><a NAME="E48:37_5"/><i><font color="Green">E147</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">ab1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">ab2</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="cat_1.html#T126">CAT_1:126</a></i>;<br/><a NAME="E49:37_5"/><i><font color="Green">E148</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">ab2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">ab2</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="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/><a NAME="E50:37_5"/><i><font color="Green">E149</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</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="cat_1.html#T52">CAT_1:52</a></i>;<br/><a NAME="E51:37_5"/><i><font color="Green">E150</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</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_2.html#E9:37"><i><font color="Green">E66</font></i></a>, <a class="ref" href="cat_1.html#T52">CAT_1:52</a></i>;<br/><a NAME="E52:37_5"/><i><font color="Green">E151</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<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> <a href="cat_1.html#K13">.</a> <font color="Maroon">a1</font></span>)</span>,<span class="p1">(<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> <a href="cat_1.html#K13">.</a> <font color="Maroon">a2</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="cat_1.html#T126">CAT_1:126</a></i>;<br/><a NAME="E53:37_5"/><i><font color="Green">E152</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a2</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="cat_1.html#T126">CAT_1:126</a></i>;<br/><a NAME="E54:37_5"/><i><font color="Green">E153</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<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> <a href="cat_1.html#K13">.</a> <font color="Maroon">a1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a1</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#T30">ISOCAT_1:30</a></i>;<br/><a NAME="E55:37_5"/><i><font color="Green">E154</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<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> <a href="cat_1.html#K13">.</a> <font color="Maroon">a2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a2</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#T30">ISOCAT_1:30</a></i>;<br/><a NAME="E56:37_5"/><i><font color="Green">E159</font></i>: 
( <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a2</font> &amp; <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a1</font> &amp; <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a1</font> &amp; <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a2</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="cat_1.html#NM2">Morphism</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> ;<br/><b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font> ;<br/><b>reconsider </b><font color="Maroon">h'</font> = <font color="Maroon">h</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">B</font> ;<br/><b>reconsider </b><font color="Maroon">e1</font> = <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font>, <font color="Maroon">e2</font> = <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</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">B</font>,<font color="Maroon">C</font></span>)</span> ;<br/><a NAME="E61:37_5"/><i><font color="Green">E160</font></i>: 
<span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g'</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">v1</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="txt" href="isocat_2.html#E5:37"><i><font color="Green">E51</font></i></a>, <a class="txt" href="isocat_2.html#E8:37"><i><font color="Green">E54</font></i></a>, <a class="ref" href="isocat_2.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E62:37_5"/><i><font color="Green">E161</font></i>: 
<span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g'</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">H1</font>,<font color="Maroon">H2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">v2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="txt" href="isocat_2.html#E5:37"><i><font color="Green">E51</font></i></a>, <a class="txt" href="isocat_2.html#E8:37"><i><font color="Green">E54</font></i></a>, <a class="ref" href="isocat_2.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E63:37_5"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a1</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">H1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a>, <a class="ref" href="isocat_2.html#T9" target="_self">Th9</a>, </i>;<br/><a NAME="E64:37_5"/>
<font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a2</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">G2</font>,<font color="Maroon">H2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a>, , </i>;<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"><font color="Maroon">G1</font>,<font color="Maroon">H2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t2</font> <a href="nattra_1.html#K8">`*`</a> <font color="Maroon">v1</font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a2</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<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> <a href="funct_2.html#K3">.</a> <font color="Maroon">g'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="nattra_1.html#T42">NATTRA_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a2</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">e1</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<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> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E11:37_5"><i><font color="Green">E74</font></i></a>, <a class="txt" href="isocat_2.html#E20:37_5"><i><font color="Green">E77</font></i></a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a1</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="nattra_1.html#D8">NATTRA_1:def 8</a></i>
<br/>.= 
<font color="Maroon">e2</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a1</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E12:37_5"><i><font color="Green">E75</font></i></a>, <a class="txt" href="isocat_2.html#E14:37_5"><i><font color="Green">E76</font></i></a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g'</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</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"><font color="Maroon">G1</font>,<font color="Maroon">H2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="nattra_1.html#K8">`*`</a> <font color="Maroon">t1</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E21:37_5"><i><font color="Green">E80</font></i></a>, <a class="txt" href="isocat_2.html#E22:37_5"><i><font color="Green">E81</font></i></a>, <a class="ref" href="nattra_1.html#T42">NATTRA_1:42</a></i>
;<br/><a NAME="E66:37_5"/><b>then </b><i><font color="Green">E163</font></i>: 
<font color="Maroon">t2</font> <a href="nattra_1.html#K8">`*`</a> <font color="Maroon">v1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">v2</font> <a href="nattra_1.html#K8">`*`</a> <font color="Maroon">t1</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><i><font color="Green">E164</font></i>: <font color="Maroon">u</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">ab2</font> = 
<font color="Maroon">u'</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t'</font> <a href="cat_2.html#K1">.</a> <font color="Maroon">a2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b2</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">t2</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E4:37"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
;<br/><a NAME="E68:37_5"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a1</font>,<font color="Maroon">a1</font> <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="cat_1.html#T56">CAT_1:56</a></i>;<br/><b>then </b><i><font color="Green">E165</font></i>: <font color="Maroon">g'</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a1</font></span>)</span> = 
<font color="Maroon">g</font> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<font color="Maroon">g</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#T58">CAT_1:58</a></i>
;<br/><a NAME="E70:37_5"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b2</font>,<font color="Maroon">b2</font> <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="cat_1.html#T56">CAT_1:56</a></i>;<br/><b>then </b><i><font color="Green">E166</font></i>: <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">h'</font> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K9">*</a> <font color="Maroon">h</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<font color="Maroon">h</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#T57">CAT_1:57</a></i>
;<br/><a NAME="E72:37_5"/><i><font color="Green">E167</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> )
 <b>by </b><i>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a>, <a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/><b>then </b><i><font color="Green">E168</font></i>:  <a href="cat_1.html#K3">cod</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a1</font></span>)</span>,<font color="Maroon">h</font></span><a href="cat_2.html#K13">]</a></span> = 
<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span>
<b>by </b><i><a class="ref" href="cat_2.html#T38">CAT_2:38</a></i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b2</font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i>, <a class="ref" href="cat_2.html#T38">CAT_2:38</a></i>
;<br/><a NAME="E74:37_5"/><b>then </b><i><font color="Green">E169</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b2</font></span>)</span></span><a href="cat_2.html#K13">]</a></span> <a href="cat_1.html#K4">*</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a1</font></span>)</span>,<font color="Maroon">h</font></span><a href="cat_2.html#K13">]</a></span>
 <b>by </b><i>, <a class="txt" href="isocat_2.html#E11"><i><font color="Green">Lemma84</font></i></a>, , <a class="ref" href="cat_2.html#T40">CAT_2:40</a></i>;<br/><a NAME="E75:37_5"/>
the <a href="cat_1.html#U2">Morphisms</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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="cat_2.html#T33">CAT_2:33</a></i>;<br/><b>then reconsider </b><font color="Maroon">FF1</font> = <font color="Maroon">F1</font>, <font color="Maroon">FF2</font> = <font color="Maroon">F2</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> ;<br/><a NAME="E77:37_5"/><i><font color="Green">E170</font></i>: 
 <a href="isocat_2.html#K3" target="_self">curry</a> <font color="Maroon">F1</font>,<font color="Maroon">g</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">FF1</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">g</font>
 ;<br/><i><font color="Green">E171</font></i>: <font color="Maroon">v1</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K3" target="_self">curry</a> <font color="Maroon">F1</font>,<font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K1">*</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">B</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b2</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K3" target="_self">curry</a> <font color="Maroon">F1</font>,<font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">B</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K3" target="_self">curry</a> <font color="Maroon">F1</font>,<font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#D5">CAT_1:def 5</a></i>
<br/>.= 
<font color="Maroon">F1</font> <a href="cat_2.html#K14">.</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_2.html#T3">CAT_2:3</a></i>
;<br/><i><font color="Green">E172</font></i>: <font color="Maroon">G1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font>
<b>by </b><i>, , <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
<font color="Maroon">F1</font> <a href="cat_2.html#K14">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a1</font></span>)</span>,<font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="cat_2.html#T47">CAT_2:47</a></i>
;<br/><i><font color="Green">E173</font></i>: <font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font> = 
<font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f'</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="cat_1.html#T99">CAT_1:99</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
;<br/><i><font color="Green">E174</font></i>: <font color="Maroon">u</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">ab1</font> = 
<font color="Maroon">u'</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a>, <a class="ref" href="isocat_2.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t'</font> <a href="cat_2.html#K1">.</a> <font color="Maroon">a1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">t1</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="isocat_2.html#D1" target="_self">Def1</a>, , <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
;<br/><a NAME="E82:37_5"/><i><font color="Green">E175</font></i>: 
 <a href="isocat_2.html#K3" target="_self">curry</a> <font color="Maroon">F2</font>,<font color="Maroon">g</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">FF2</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">g</font>
 ;<br/><i><font color="Green">E176</font></i>: <font color="Maroon">v2</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K3" target="_self">curry</a> <font color="Maroon">F2</font>,<font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K1">*</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">B</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b2</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K3" target="_self">curry</a> <font color="Maroon">F2</font>,<font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">B</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K3" target="_self">curry</a> <font color="Maroon">F2</font>,<font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#D5">CAT_1:def 5</a></i>
<br/>.= 
<font color="Maroon">F2</font> <a href="cat_2.html#K14">.</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b2</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E31:37_5"><i><font color="Green">E94</font></i></a>, <a class="ref" href="cat_2.html#T3">CAT_2:3</a></i>
;<br/><i><font color="Green">E180</font></i>: <font color="Maroon">H1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font>
<b>by </b><i>, , <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
<font color="Maroon">F2</font> <a href="cat_2.html#K14">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a1</font></span>)</span>,<font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="cat_2.html#T47">CAT_2:47</a></i>
;<br/><i><font color="Green">E181</font></i>: <font color="Maroon">F2</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font> = 
<font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f'</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="isocat_2.html#E32:37_5"><i><font color="Green">E95</font></i></a>, <a class="txt" href="isocat_2.html#E33:37_5"><i><font color="Green">E96</font></i></a>, <a class="ref" href="cat_1.html#T99">CAT_1:99</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i>, <a class="txt" href="isocat_2.html#E9:37"><i><font color="Green">E66</font></i></a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">ab2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">t2</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E9:37_5"><i><font color="Green">E70</font></i></a>, <a class="txt" href="isocat_2.html#E10:37_5"><i><font color="Green">E71</font></i></a>, <a class="txt" href="isocat_2.html#E26:37_5"><i><font color="Green">E83</font></i></a>, <a class="txt" href="isocat_2.html#E27:37_5"><i><font color="Green">E92</font></i></a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t2</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">h</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t2</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i>, , , <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">v2</font> <a href="nattra_1.html#K8">`*`</a> <font color="Maroon">t1</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="isocat_2.html#E23:37_5"><i><font color="Green">E82</font></i></a>, <a class="ref" href="nattra_1.html#T27">NATTRA_1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="nattra_1.html#T27">NATTRA_1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">h</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="cat_1.html#T54">CAT_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">h</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="nattra_1.html#D8">NATTRA_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i>, <a class="txt" href="isocat_2.html#E9:37"><i><font color="Green">E66</font></i></a>, <a class="txt" href="isocat_2.html#E12:37"><i><font color="Green">E67</font></i></a>, <a class="ref" href="cat_1.html#T54">CAT_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">ab1</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E12:37"><i><font color="Green">E67</font></i></a>, , <a class="txt" href="isocat_2.html#E30:37_5"><i><font color="Green">E93</font></i></a>, <a class="txt" href="isocat_2.html#E34:37_5"><i><font color="Green">E97</font></i></a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F2</font> <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">u</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">ab1</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E1:37_5"><i><font color="Green">E68</font></i></a>, <a class="txt" href="isocat_2.html#E3:37_5"><i><font color="Green">E69</font></i></a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E13:37"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:37_6"/>
<font color="Maroon">F1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">F2</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a><br/>

<b>thus </b><a NAME="E2:37_6"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="nattra_1.html#M1">transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font> st <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</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>  holds <br/>(  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or for <font color="Olive">b<sub>4</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> holds  <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="nattra_1.html#K3">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )
 <b>by </b><i/>;<br/>


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

<a NAME="E14:37"/>
<font color="Maroon">u</font> is    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_7"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:37_7"/>
<font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D8">NATTRA_1:def 8</a><br/>

<b>thus </b><a NAME="E2:37_7"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</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>  holds <br/>(  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds  <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="nattra_1.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font> ;<br/>
<b>take </b>
<font color="Maroon">u</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>;<br/><b>assume </b><a NAME="E1:37_8"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">u</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/><a NAME="E2:37_8"/>
<font color="Maroon">t'</font> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K3">curry</a> <font color="Maroon">u'</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:37_8"/>
<font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">s</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a>, <a class="ref" href="isocat_2.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:37"/>
<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/>


</div>
