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<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">F1</font>, <font color="Olive">F2</font> being   <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font><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">a<sub>1</sub></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"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Olive">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Olive">F2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" target="_self">export</a> <font color="Olive">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/>
<div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:38_1_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font></span>)</span>
 ;<br/><a NAME="E2:38_1_1_1"/><b>then </b>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11">NatTrans</a> <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/><b>then consider </b><font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>, <font color="Maroon">t</font> being    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E4:38_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">o</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>and </b><br/><a NAME="E5:38_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D16">NATTRA_1:def 16</a></i>;<br/><b>take </b><font color="Maroon">m</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" target="_self">export</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/><a NAME="E6:38_1_1_1"/>
 <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a>, <a class="ref" href="isocat_2.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E7:38_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11">NatTrans</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="nattra_1.html#D16">NATTRA_1:def 16</a></i>;<br/><b>hence </b><a NAME="E8:38_1_1_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">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/><b>thus </b><a NAME="E9:38_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">o</font>,<font color="Maroon">m</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F1'</font> be    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>, <font color="Maroon">F2'</font> be    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>;<br/><b>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="E1:38_1_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">o</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="E2:38_1_1_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">t</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">t'</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E3:38_1_1_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="E4:38_1_1_1_1"/>
<font color="Maroon">m</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"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1'</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" target="_self">export</a> <font color="Maroon">t'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <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">FF</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font></span>)</span>,the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:38_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">o</font>,<font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<a NAME="E4:38_1_1"/>
<font color="Maroon">FF</font> is    <a href="cat_1.html#M2">Functor</a> of  <a href="nattra_1.html#K12">Functors</a> <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>, <a href="nattra_1.html#K12">Functors</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:38_1_1_2"/>
for <font color="Olive">c</font> being  <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font></span>)</span>  ex <font color="Olive">d</font> being  <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span></span>)</span> st <font color="Maroon">FF</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">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Olive">d</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="38_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">c</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> <b>by </b><i/>;<br/>
<a NAME="E2:38_1_1_2_1"/><i><font color="Green">E51</font></i>: 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c</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">F</font>,<font color="Maroon">F</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">F</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>reconsider </b><font color="Maroon">d</font> =  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F</font> as   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">d</font>
;<br/>

<a NAME="E4:38_1_1_2_1"/><i><font color="Green">E54</font></i>: 
 <a href="nattra_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="isocat_2.html#K6" target="_self">export</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">FF</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">c</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" target="_self">export</a> <span class="p3">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">F</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="isocat_2.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">d</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>
;<br/>


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

<b>thus </b><a NAME="E2:38_1_1_2"/>
for <font color="Olive">f</font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font></span>)</span> holds <br/> ( <font color="Maroon">FF</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">FF</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> &amp; <font color="Maroon">FF</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">FF</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="38_1_1_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">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>, <font color="Maroon">t</font> being    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E2:38_1_1_2_2"/>
<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="E3:38_1_1_2_2"/><i><font color="Green">E55</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E4:38_1_1_2_2"/><i><font color="Green">E57</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="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">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#T6" target="_self">Th6</a>, <a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E5:38_1_1_2_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">FF</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"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" target="_self">export</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="ref" href="isocat_2.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E6:38_1_1_2_2"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#T39">NATTRA_1:39</a></i>;<br/>
<b>thus </b><font color="Maroon">FF</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"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" target="_self">export</a> <span class="p3">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">F1</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><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">FF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>
;<br/>

<a NAME="E8:38_1_1_2_2"/><i><font color="Green">E60</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="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">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="isocat_2.html#T6" target="_self">Th6</a>, <a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E9:38_1_1_2_2"/><i><font color="Green">E61</font></i>: 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">FF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font>
 
<b>by </b><i><a class="ref" href="isocat_2.html#T7" target="_self">Th7</a>, <a class="ref" href="nattra_1.html#T39">NATTRA_1:39</a></i>;<br/>
<b>thus </b><font color="Maroon">FF</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"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" target="_self">export</a> <span class="p3">(<span class="default"><a href="nattra_1.html#K7">id</a> <font color="Maroon">F2</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7">id</a> <span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><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">FF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>
;<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">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font></span>)</span>, <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>, <font color="Maroon">t</font> being    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E4:38_1_1_2"/><i><font color="Green">E62</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="E5:38_1_1_2"/><i><font color="Green">E65</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E6:38_1_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">FF</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"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" target="_self">export</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">G1</font> being    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>, <font color="Maroon">G2</font> being    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>, <font color="Maroon">u</font> being    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">G1</font>,<font color="Maroon">G2</font><b> such that </b><br/><a NAME="E8:38_1_1_2"/><i><font color="Green">E67</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="E9:38_1_1_2"/><i><font color="Green">E68</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E10:38_1_1_2"/><i><font color="Green">E69</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>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F2</font>,<font color="Maroon">G2</font> <b>by </b><i>, </i>;<br/>
<a NAME="E12:38_1_1_2"/>
<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="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">u</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="nattra_1.html#T42">NATTRA_1:42</a></i>;<br/>
<a NAME="E13:38_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">FF</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> <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"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">G2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" target="_self">export</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="nattra_1.html#K8">`*`</a> <font color="Maroon">t</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E14:38_1_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">FF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">G2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" target="_self">export</a> <font color="Maroon">u</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E15:38_1_1_2"/>
<span class="p1">(<span class="default"><a href="isocat_2.html#K6" target="_self">export</a> <font color="Maroon">u</font></span>)</span> <a href="nattra_1.html#K8">`*`</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K6" target="_self">export</a> <font color="Maroon">t</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="isocat_2.html#K6" target="_self">export</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nattra_1.html#K8">`*`</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a>, , , , , </i>;<br/>
<b>hence </b><a NAME="E16:38_1_1_2"/>
<font color="Maroon">FF</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">FF</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">FF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
 <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="isocat_2.html#T9" target="_self">Th9</a>, <a class="ref" href="nattra_1.html#T42">NATTRA_1:42</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">FF</font> = <font color="Maroon">FF</font> as    <a href="cat_1.html#M2">Functor</a> of  <a href="nattra_1.html#K12">Functors</a> <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>, <a href="nattra_1.html#K12">Functors</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">FF</font>
;<br/>

<b>let </b><font color="Maroon">F1</font> be    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>, <font color="Maroon">F2</font> be    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E6:38_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 ;<br/>

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

<a NAME="E7:38_1_1"/>
<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> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11">NatTrans</a> <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a>, <a class="ref" href="nattra_1.html#T35">NATTRA_1:35</a></i>;<br/>
<a NAME="E8:38_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">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> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/>
<b>hence </b><a NAME="E9:38_1_1"/>
<font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">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> <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"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" target="_self">export</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>


</div>
