<?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">G</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>, <a href="nattra_1.html#K12">Functors</a> <font color="Maroon">B</font>,<font color="Maroon">C</font>;<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">f</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font><br/> for <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">B</font>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">f</font>,<font color="Olive">g</font></span><a href="cat_2.html#K13">]</a></span> holds <br/>for <font color="Olive">f1</font>, <font color="Olive">f2</font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font><br/> for <font color="Olive">t</font> being   <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Olive">f1</font>,<font color="Olive">f2</font>  st <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">f1</font>,<font color="Olive">f2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">t</font></span><a href="tarski.html#K4">]</a></span> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Olive">g</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Olive">f1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font></span>)</span>;<br/>
<div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1"><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>then consider </b><font color="Maroon">m1</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font>, <font color="Maroon">m2</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E3:36_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">m1</font>,<font color="Maroon">m2</font></span><a href="cat_2.html#K13">]</a></span>
 <b>by </b><i><a class="ref" href="cat_2.html#T37">CAT_2:37</a></i>;<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">t1</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="E5:36_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="E6:36_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">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m1</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">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m1</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">t1</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">o</font> = <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">m2</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m2</font></span>)</span> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">o</font> = <font color="Maroon">o</font>;<br/><b>thus </b><a NAME="E8:36_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>
 ;<br/><b>thus </b><a NAME="E9:36_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">o</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">B</font>;<br/>



<b>assume </b><a NAME="E1:36_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_2.html#K13">]</a></span>
 ;<br/>

<a NAME="E2:36_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m2</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">f1</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>, <font color="Maroon">f2</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">t</font> be    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">f1</font>,<font color="Maroon">f2</font>;<br/>



<b>assume </b><a NAME="E3:36_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<a NAME="E4:36_1_1"/><b>then </b>
( <span class="p1"><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> <a href="hidden.html#R1">=</a> <span class="p1"><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> &amp; <font color="Maroon">t1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">t</font> )
 
<b>by </b><i>, <a class="ref" href="isocat_2.html#T6" target="_self">Th6</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E5:36_1_1"/><b>then </b>
( <font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">F2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f2</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E6:36_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="isocat_2.html#T6" target="_self">Th6</a>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of 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>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:36"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">m</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a> 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> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<a NAME="E4:36"/>
<font color="Maroon">F</font> is    <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:36_2"/>
for <font color="Olive">ab</font> being  <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>  ex <font color="Olive">c</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> st <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">ab</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Olive">c</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="isocat_1.html#D1">ISOCAT_1:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1"><b>proof </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="E2:36_2_1"/><i><font color="Green">E54</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="cat_2.html#T35">CAT_2:35</a></i>;<br/>
<a NAME="E3:36_2_1"/><i><font color="Green">E55</font></i>: 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">ab</font> <a href="hidden.html#R1">=</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">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b</font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
 
<b>by </b><i>, <a class="ref" href="cat_2.html#T41">CAT_2:41</a></i>;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">H</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font>
;<br/>

<a NAME="E5:36_2_1"/><i><font color="Green">E57</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H</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="cat_1.html#T56">CAT_1:56</a></i>;<br/>
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span> = 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T108">CAT_1:108</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">H</font>,<font color="Maroon">H</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">H</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>
;<br/>
<b>hence </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">ab</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">H</font></span>)</span> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="isocat_2.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">H</font></span>)</span> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T108">CAT_1:108</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">H</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nattra_1.html#T21">NATTRA_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T59">CAT_1:59</a></i>
;<br/><br/>


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

<b>thus </b><a NAME="E2:36_2"/>
for <font color="Olive">f</font> being  <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> holds <br/> ( <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Olive">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Olive">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <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>consider </b><font color="Maroon">f1</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font>, <font color="Maroon">f2</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E2:36_2_2"/><i><font color="Green">E58</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">f1</font>,<font color="Maroon">f2</font></span><a href="cat_2.html#K13">]</a></span>
 <b>by </b><i><a class="ref" href="cat_2.html#T37">CAT_2:37</a></i>;<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">t</font> being    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E4:36_2_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 <b>and </b><br/><a NAME="E5:36_2_2"/><i><font color="Green">E60</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f1</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">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<i><font color="Green">E61</font></i>:  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> = 
 <a href="cat_1.html#K10">id</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f1</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font></span>)</span></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/>.= 
<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> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="ref" href="cat_2.html#T41">CAT_2:41</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f1</font></span>)</span> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font> <b>by </b><i/>;<br/>
<a NAME="E8:36_2_2"/><i><font color="Green">E62</font></i>: 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f1</font></span>)</span></span>)</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">H</font>,<font color="Maroon">H</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">H</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E9:36_2_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T109">CAT_1:109</a></i>;<br/>
<a NAME="E10:36_2_2"/><i><font color="Green">E66</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> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></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#T6" target="_self">Th6</a>, <a class="ref" href="isocat_1.html#T30">ISOCAT_1:30</a></i>;<br/>
<a NAME="E11:36_2_2"/>
<font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T109">CAT_1:109</a></i>;<br/>
<a NAME="E12:36_2_2"/><b>then </b><i><font color="Green">E67</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> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<a NAME="E13:36_2_2"/><i><font color="Green">E68</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></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#T56">CAT_1:56</a></i>;<br/>
<a NAME="E14:36_2_2"/>
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T108">CAT_1:108</a></i>;<br/>
<b>hence </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">H</font></span>)</span> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="isocat_2.html#T7" target="_self">Th7</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">H</font></span>)</span> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="cat_1.html#K13">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T108">CAT_1:108</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">H</font></span>)</span> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T109">CAT_1:109</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">H</font></span>)</span> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="cat_1.html#K13">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nattra_1.html#T21">NATTRA_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T109">CAT_1:109</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="isocat_2.html#T9" target="_self">Th9</a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#T59">CAT_1:59</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="isocat_2.html#T8" target="_self">Th8</a>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
;<br/><br/>

<i><font color="Green">E69</font></i>:  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span> = 
 <a href="cat_1.html#K10">id</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f1</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></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/>.= 
<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> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="ref" href="cat_2.html#T41">CAT_2:41</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f1</font></span>)</span> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font> <b>by </b><i/>;<br/>
<a NAME="E18:36_2_2"/><i><font color="Green">E70</font></i>: 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f1</font></span>)</span></span>)</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">H</font>,<font color="Maroon">H</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">H</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E19:36_2_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">F2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T109">CAT_1:109</a></i>;<br/>
<a NAME="E20:36_2_2"/><i><font color="Green">E74</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> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></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#T6" target="_self">Th6</a>, <a class="ref" href="isocat_1.html#T30">ISOCAT_1:30</a></i>;<br/>
<a NAME="E21:36_2_2"/>
<font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T109">CAT_1:109</a></i>;<br/>
<a NAME="E22:36_2_2"/><b>then </b><i><font color="Green">E75</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> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/>
<a NAME="E23:36_2_2"/><i><font color="Green">E76</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></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#T56">CAT_1:56</a></i>;<br/>
<a NAME="E24:36_2_2"/>
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="cat_1.html#T108">CAT_1:108</a></i>;<br/>
<b>hence </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">H</font></span>)</span> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="isocat_2.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">H</font></span>)</span> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="cat_1.html#K13">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T108">CAT_1:108</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">H</font></span>)</span> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T109">CAT_1:109</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">H</font></span>)</span> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="cat_1.html#K13">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nattra_1.html#T21">NATTRA_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T109">CAT_1:109</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T59">CAT_1:59</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#T109">CAT_1:109</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
;<br/><br/>


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

<b>let </b><font color="Maroon">f</font> be   <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>, <font color="Maroon">g</font> be   <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>assume </b><a NAME="E3:36_2"/><i><font color="Green">E77</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
 ;<br/>

<b>consider </b><font color="Maroon">f1</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font>, <font color="Maroon">f2</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E5:36_2"/><i><font color="Green">E80</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">f1</font>,<font color="Maroon">f2</font></span><a href="cat_2.html#K13">]</a></span>
 <b>by </b><i><a class="ref" href="cat_2.html#T37">CAT_2:37</a></i>;<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">t</font> being    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E7:36_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 <b>and </b><br/><a NAME="E8:36_2"/><i><font color="Green">E82</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f1</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">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">g1</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font>, <font color="Maroon">g2</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E10:36_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">g2</font></span><a href="cat_2.html#K13">]</a></span>
 <b>by </b><i><a class="ref" href="cat_2.html#T37">CAT_2:37</a></i>;<br/>
<b>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">s</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="E12:36_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">G1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G2</font>
 <b>and </b><br/><a NAME="E13:36_2"/><i><font color="Green">E93</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> &amp; <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</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">s</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E14:36_2"/>
( <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f1</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span><a href="cat_2.html#K12">]</a></span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g1</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</font></span>)</span></span><a href="cat_2.html#K12">]</a></span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> )
 
<b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a>, , <a class="ref" href="cat_2.html#T38">CAT_2:38</a></i>;<br/>
<a NAME="E15:36_2"/><b>then </b><i><font color="Green">E94</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g1</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E16:36_2"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f2</font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
 
<b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a>, , <a class="ref" href="cat_2.html#T39">CAT_2:39</a></i>;<br/>
<a NAME="E17:36_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">F2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="isocat_2.html#T7" target="_self">Th7</a>, , <a class="ref" href="isocat_2.html#T8" target="_self">Th8</a>, <a class="ref" href="cat_1.html#T99">CAT_1:99</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F2</font>,<font color="Maroon">G2</font> <b>by </b><i><a class="ref" href="isocat_2.html#T7" target="_self">Th7</a>, , <a class="ref" href="isocat_2.html#T8" target="_self">Th8</a>, <a class="ref" href="cat_1.html#T99">CAT_1:99</a></i>;<br/>
<i><font color="Green">E97</font></i>: <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f1</font></span>)</span>
<b>by </b><i><a class="ref" href="isocat_2.html#T8" target="_self">Th8</a>, <a class="ref" href="cat_1.html#T99">CAT_1:99</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">F1</font>,<font color="Maroon">G2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K8">`*`</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="isocat_2.html#T7" target="_self">Th7</a>, , , <a class="ref" href="nattra_1.html#T42">NATTRA_1:42</a></i>
;<br/>
<a NAME="E20:36_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="isocat_2.html#T6" target="_self">Th6</a>, <a class="ref" href="isocat_2.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E21:36_2"/><i><font color="Green">E99</font></i>: 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font>
 
<b>by </b><i><a class="ref" href="isocat_2.html#T8" target="_self">Th8</a>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/>
<a NAME="E22:36_2"/><i><font color="Green">E100</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> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></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="E23:36_2"/><i><font color="Green">E101</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> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></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="E24:36_2"/><i><font color="Green">E138</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</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#T8" target="_self">Th8</a>, <a class="ref" href="cat_1.html#T19">CAT_1:19</a></i>;<br/>
<a NAME="E25:36_2"/><b>then </b><i><font color="Green">E139</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> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</font></span>)</span></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="E26:36_2"/><i><font color="Green">E140</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</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#T19">CAT_1:19</a></i>;<br/>
<a NAME="E27:36_2"/><b>then </b><i><font color="Green">E141</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> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></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="E28:36_2"/><b>then </b><i><font color="Green">E142</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> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></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#E3:36"><i><font color="Green">E51</font></i></a>, <a class="ref" href="cat_1.html#T52">CAT_1:52</a></i>;<br/>
<a NAME="E29:36_2"/><i><font color="Green">E143</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> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></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="E30:36_2"/><i><font color="Green">E146</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> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</font></span>)</span></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="E31:36_2"/><b>then </b><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> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</font></span>)</span></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#E3:36"><i><font color="Green">E51</font></i></a>, <a class="ref" href="cat_1.html#T52">CAT_1:52</a></i>;<br/>
<a NAME="E32:36_2"/><i><font color="Green">E148</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> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></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="E33:36_2"/><i><font color="Green">E149</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> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></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/>
<b>reconsider </b><font color="Maroon">f2'</font> = <font color="Maroon">f2</font> as    <a href="cat_1.html#M1">Morphism</a> of  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font>, <a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</font> <b>by </b><i><a class="ref" href="isocat_2.html#T8" target="_self">Th8</a>, <a class="ref" href="cat_1.html#T22">CAT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g2'</font> = <font color="Maroon">g2</font> as    <a href="cat_1.html#M1">Morphism</a> of  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</font>, <a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font> <b>by </b><i><a class="ref" href="cat_1.html#T22">CAT_1:22</a></i>;<br/>
<a NAME="E36:36_2"/><i><font color="Green">E150</font></i>: 
( <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g2'</font> &amp; <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f2'</font> &amp; <font color="Maroon">G1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g2'</font> )
 
<b>by </b><i>, , , <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>;<br/>
<b>thus </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K8">`*`</a> <font color="Maroon">t</font></span>)</span> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><font color="Maroon">g2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="isocat_2.html#T9" target="_self">Th9</a>, <a class="ref" href="isocat_2.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f2</font></span>)</span></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"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g2</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="isocat_2.html#T8" target="_self">Th8</a>, <a class="ref" href="cat_1.html#T99">CAT_1:99</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g2'</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f2'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:36"><i><font color="Green">E51</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"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g2'</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f2'</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"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g2'</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f2'</font></span>)</span></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">s</font> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g2'</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f2'</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="isocat_2.html#E3:36"><i><font color="Green">E51</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">s</font> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F2</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g2'</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f2'</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"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g2'</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f2'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:36"><i><font color="Green">E51</font></i></a>, , , <a class="ref" href="cat_1.html#T54">CAT_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g2'</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f2'</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"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g2'</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f2'</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">s</font> <a href="nattra_1.html#K5">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g2'</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="isocat_2.html#T8" target="_self">Th8</a>, <a class="txt" href="isocat_2.html#E3:36"><i><font color="Green">E51</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"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2</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"><font color="Maroon">F</font> <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">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, , , , </i>
;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as    <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>take </b>
<font color="Maroon">F</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</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">t</font> being    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">f1</font>,<font color="Maroon">f2</font><b> such that </b><br/><a NAME="E2:36_3"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">f1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">f2</font>
 <b>and </b><br/><a NAME="E3:36_3"/><i><font color="Green">E152</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</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">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> )
 <b>and </b><br/><a NAME="E4:36_3"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">B</font>;<br/><a NAME="E1:36_3_1"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a>, <a class="ref" href="cat_1.html#T109">CAT_1:109</a></i>;<br/><i><font color="Green">E159</font></i>: <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T108">CAT_1:108</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">f1</font>,<font color="Maroon">f1</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">f1</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="isocat_2.html#T7" target="_self">Th7</a>, <a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>
;<br/><a NAME="E3:36_3_1"/><i><font color="Green">E160</font></i>: 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_2.html#K15">?-</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> = 
<font color="Maroon">F</font> <a href="cat_2.html#K14">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="cat_2.html#T47">CAT_2:47</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</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> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span><a href="cat_2.html#K13">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">f1</font></span>)</span> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="cat_1.html#K13">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <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#T21">NATTRA_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T108">CAT_1:108</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#T99">CAT_1:99</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="cat_1.html#T46">CAT_1:46</a></i>
;<br/></div><b>end;</b></div><a NAME="E6:36_3"/><b>then </b><i><font color="Green">E161</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F</font> <a href="cat_2.html#K15">?-</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">B</font>;<br/><a NAME="E1:36_3_2"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a>, <a class="ref" href="cat_1.html#T109">CAT_1:109</a></i>;<br/><i><font color="Green">E163</font></i>: <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T108">CAT_1:108</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">f2</font>,<font color="Maroon">f2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">f2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>
;<br/><a NAME="E3:36_3_2"/><i><font color="Green">E164</font></i>: 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_2.html#K15">?-</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> = 
<font color="Maroon">F</font> <a href="cat_2.html#K14">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="cat_2.html#T47">CAT_2:47</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</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> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span><a href="cat_2.html#K13">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">f2</font></span>)</span> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="cat_1.html#K13">.</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <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#T21">NATTRA_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T108">CAT_1:108</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#T99">CAT_1:99</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="cat_1.html#T46">CAT_1:46</a></i>
;<br/></div><b>end;</b></div><a NAME="E8:36_3"/><b>then </b><i><font color="Green">E165</font></i>: 
<font color="Maroon">f2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F</font> <a href="cat_2.html#K15">?-</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">B</font>;<br/><a NAME="E1:36_3_3"/><i><font color="Green">E166</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">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f2</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/><a NAME="E2:36_3_3"/><i><font color="Green">E167</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">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f1</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="cat_1.html#T56">CAT_1:56</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="isocat_2.html#K4" target="_self">?-</a> <font color="Maroon">f</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b</font> = 
<font color="Maroon">F</font> <a href="cat_2.html#K14">.</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="isocat_2.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b</font></span>)</span></span><a href="cat_2.html#K13">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T108">CAT_1:108</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="cat_1.html#D13">CAT_1:def 13</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="cat_1.html#T58">CAT_1:58</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:36_3"/>
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="cat_2.html#K15">?-</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="cat_2.html#K15">?-</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="isocat_2.html#K4" target="_self">?-</a> <font color="Maroon">f</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="ref" href="isocat_2.html#T7" target="_self">Th7</a>, , <a class="ref" href="isocat_1.html#T31">ISOCAT_1:31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:36"/>
<font color="Maroon">G</font> <a href="funct_2.html#R2">=</a>  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="isocat_2.html#T3" target="_self">Th3</a></i>;<br/>


</div>
