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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font><br/> for <font color="Olive" title="b3">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Olive" title="b1">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Olive" title="b2">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Olive" title="b3">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<div><i><font color="Green" title="E33">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_1_1_1"/>
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font></span>)</span>
 ;<br/><a NAME="E2:37_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">F1</font>, <font color="Maroon" title="c6">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font>, <font color="Maroon" title="c7">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font><b> such that </b><br/><a NAME="E4:37_1_1_1"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c7">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E5:37_1_1_1"/><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c5">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">F2</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D16" title="NATTRA_1:def.16">NATTRA_1:def 16</a></i>;<br/><b>take </b><font color="Maroon" title="c8">m</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c6">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Maroon" title="c7">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/><a NAME="E6:37_1_1_1"/>
 <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c5">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c6">F2</font>
 <b>by </b><i><a class="txt" href="isocat_2.html#E5:37_1_1_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="isocat_2.html#T27" target="_self" title="ISOCAT_2:th.27">Th27</a></i>;<br/><a NAME="E7:37_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="nattra_1.html#D16" title="NATTRA_1:def.16">NATTRA_1:def 16</a></i>;<br/><b>hence </b><a NAME="E8:37_1_1_1"/>
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/><b>thus </b><a NAME="E9:37_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c8">m</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">F1'</font>, <font color="Maroon" title="c10">F2'</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font>;<br/><b>let </b><font color="Maroon" title="c11">t'</font> be    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c9">F1'</font>,<font color="Maroon" title="c10">F2'</font>;<br/>



<b>assume </b><a NAME="E1:37_1_1_1_1"/><i><font color="Green" title="E36">A4</font></i>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1'</font>,<font color="Maroon" title="c10">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">t'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/>

<a NAME="E2:37_1_1_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1'</font>,<font color="Maroon" title="c10">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c7">t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">t'</font> )
 
<b>by </b><i><a class="txt" href="isocat_2.html#E4:37_1_1_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E3:37_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c5">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">F1'</font> &amp; <font color="Maroon" title="c6">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c10">F2'</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E4:37_1_1_1_1"/>
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c9">F1'</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c10">F2'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Maroon" title="c11">t'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="isocat_2.html#E4:37_1_1_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="isocat_2.html#E1:37_1_1_1_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">FF</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font></span>)</span>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:37_1_1"/><i><font color="Green" title="E34">A5</font></i>: 
for <font color="Olive" title="b1">o</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c4">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="isocat_2.html#E1:37_1_1"><i><font color="Green" title="E33">A1</font></i></a>);<br/></i>
<a NAME="E4:37_1_1"/>
<font color="Maroon" title="c4">FF</font> is    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of  <a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font>, <a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:37_1_1_2"/>
for <font color="Olive" title="b1">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font></span>)</span>  ex <font color="Olive" title="b2">d</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span></span>)</span> st <font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Olive" title="b2">d</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="isocat_1.html#D1" title="ISOCAT_1:def.1">ISOCAT_1:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">c</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">F</font> = <font color="Maroon" title="c5">c</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font> <b>by </b><i><a class="ref" href="isocat_2.html#T8" target="_self" title="ISOCAT_2:th.8">Th8</a></i>;<br/>
<a NAME="E2:37_1_1_2_1"/><i><font color="Green" title="E35">A6</font></i>: 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c6">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c6">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">d</font> =  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c6">F</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="isocat_2.html#T8" target="_self" title="ISOCAT_2:th.8">Th8</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">d</font>
;<br/>

<a NAME="E4:37_1_1_2_1"/><i><font color="Green" title="E36">A7</font></i>: 
 <a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <span class="p1">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c6">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="isocat_2.html#T28" target="_self" title="ISOCAT_2:th.28">Th28</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c5">c</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c6">F</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c6">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <span class="p3">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c6">F</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:37_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="isocat_2.html#E2:37_1_1_2_1"><i><font color="Green" title="E35">A6</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c7">d</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E4:37_1_1_2_1"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>
;<br/>


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

<b>thus </b><a NAME="E2:37_1_1_2"/>
for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font></span>)</span> holds <br/> ( <font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon" title="c6">F1</font>, <font color="Maroon" title="c7">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font>, <font color="Maroon" title="c8">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font><b> such that </b><br/><a NAME="E2:37_1_1_2_2"/>
<font color="Maroon" title="c6">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">F2</font>
 <b>and </b><br/><a NAME="E3:37_1_1_2_2"/><i><font color="Green" title="E35">A8</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F1</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F2</font> &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c8">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="isocat_2.html#T9" target="_self" title="ISOCAT_2:th.9">Th9</a></i>;<br/>
<a NAME="E4:37_1_1_2_2"/><i><font color="Green" title="E36">A9</font></i>: 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c6">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c6">F1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E3:37_1_1_2_2"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E5:37_1_1_2_2"/><i><font color="Green" title="E37">A10</font></i>: 
<font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c6">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c7">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Maroon" title="c8">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E3:37_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="isocat_2.html#E3:37_1_1_2_2"><i><font color="Green" title="E35">A8</font></i></a></i>;<br/>
<a NAME="E6:37_1_1_2_2"/><b>then </b><i><font color="Green" title="E38">A11</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c6">F1</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#T39" title="NATTRA_1:th.39">NATTRA_1:39</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c6">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c6">F1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <span class="p3">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c6">F1</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:37_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="isocat_2.html#E4:37_1_1_2_2"><i><font color="Green" title="E36">A9</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c6">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c6">F1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c6">F1</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="isocat_2.html#T28" target="_self" title="ISOCAT_2:th.28">Th28</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E6:37_1_1_2_2"><i><font color="Green" title="E38">A11</font></i></a>, <a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>
;<br/>

<a NAME="E8:37_1_1_2_2"/><i><font color="Green" title="E39">A12</font></i>: 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">F2</font>,<font color="Maroon" title="c7">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c7">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E3:37_1_1_2_2"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<a NAME="E9:37_1_1_2_2"/><i><font color="Green" title="E40">A13</font></i>: 
 <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c7">F2</font>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E5:37_1_1_2_2"><i><font color="Green" title="E37">A10</font></i></a>, <a class="ref" href="nattra_1.html#T39" title="NATTRA_1:th.39">NATTRA_1:39</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c7">F2</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c7">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <span class="p3">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c7">F2</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:37_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="isocat_2.html#E8:37_1_1_2_2"><i><font color="Green" title="E39">A12</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c7">F2</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c7">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c7">F2</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="isocat_2.html#T28" target="_self" title="ISOCAT_2:th.28">Th28</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E9:37_1_1_2_2"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>
;<br/>


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

<b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon" title="c7">F1</font>, <font color="Maroon" title="c8">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font>, <font color="Maroon" title="c9">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">F1</font>,<font color="Maroon" title="c8">F2</font><b> such that </b><br/><a NAME="E4:37_1_1_2"/><i><font color="Green" title="E35">A14</font></i>: 
<font color="Maroon" title="c7">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">F2</font>
 <b>and </b><br/><a NAME="E5:37_1_1_2"/><i><font color="Green" title="E36">A15</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F1</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F2</font> &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">F1</font>,<font color="Maroon" title="c8">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c9">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="isocat_2.html#T9" target="_self" title="ISOCAT_2:th.9">Th9</a></i>;<br/>
<a NAME="E6:37_1_1_2"/><i><font color="Green" title="E37">A16</font></i>: 
<font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c8">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Maroon" title="c9">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E3:37_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="isocat_2.html#E5:37_1_1_2"><i><font color="Green" title="E36">A15</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">G1</font>, <font color="Maroon" title="c11">G2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font>, <font color="Maroon" title="c12">u</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">G1</font>,<font color="Maroon" title="c11">G2</font><b> such that </b><br/><a NAME="E8:37_1_1_2"/><i><font color="Green" title="E38">A17</font></i>: 
<font color="Maroon" title="c10">G1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">G2</font>
 <b>and </b><br/><a NAME="E9:37_1_1_2"/><i><font color="Green" title="E39">A18</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">G1</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">G2</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">G1</font>,<font color="Maroon" title="c11">G2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="isocat_2.html#T9" target="_self" title="ISOCAT_2:th.9">Th9</a></i>;<br/>
<b>assume </b><a NAME="E10:37_1_1_2"/><i><font color="Green" title="E40">A19</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c13">u</font> = <font color="Maroon" title="c12">u</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c8">F2</font>,<font color="Maroon" title="c11">G2</font> <b>by </b><i><a class="txt" href="isocat_2.html#E5:37_1_1_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="txt" href="isocat_2.html#E9:37_1_1_2"><i><font color="Green" title="E39">A18</font></i></a></i>;<br/>
<a NAME="E12:37_1_1_2"/>
<font color="Maroon" title="c6">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">F1</font>,<font color="Maroon" title="c11">G2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">u</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c9">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E5:37_1_1_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="txt" href="isocat_2.html#E9:37_1_1_2"><i><font color="Green" title="E39">A18</font></i></a>, <a class="txt" href="isocat_2.html#E10:37_1_1_2"><i><font color="Green" title="E40">A19</font></i></a>, <a class="ref" href="nattra_1.html#T42" title="NATTRA_1:th.42">NATTRA_1:42</a></i>;<br/>
<a NAME="E13:37_1_1_2"/><b>then </b><i><font color="Green" title="E41">A20</font></i>: 
<font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c11">G2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">u</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c9">t</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E3:37_1_1"><i><font color="Green" title="E34">A5</font></i></a></i>;<br/>
<a NAME="E14:37_1_1_2"/><i><font color="Green" title="E42">A21</font></i>: 
<font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c8">F2</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c11">G2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Maroon" title="c13">u</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E3:37_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="isocat_2.html#E5:37_1_1_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="txt" href="isocat_2.html#E9:37_1_1_2"><i><font color="Green" title="E39">A18</font></i></a>, <a class="txt" href="isocat_2.html#E10:37_1_1_2"><i><font color="Green" title="E40">A19</font></i></a></i>;<br/>
<a NAME="E15:37_1_1_2"/>
<span class="p1">(<span class="default"><a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Maroon" title="c13">u</font></span>)</span> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Maroon" title="c9">t</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">u</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c9">t</font></span>)</span>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E4:37_1_1_2"><i><font color="Green" title="E35">A14</font></i></a>, <a class="txt" href="isocat_2.html#E5:37_1_1_2"><i><font color="Green" title="E36">A15</font></i></a>, <a class="txt" href="isocat_2.html#E8:37_1_1_2"><i><font color="Green" title="E38">A17</font></i></a>, <a class="txt" href="isocat_2.html#E9:37_1_1_2"><i><font color="Green" title="E39">A18</font></i></a>, <a class="txt" href="isocat_2.html#E10:37_1_1_2"><i><font color="Green" title="E40">A19</font></i></a>, <a class="ref" href="isocat_2.html#T29" target="_self" title="ISOCAT_2:th.29">Th29</a></i>;<br/>
<b>hence </b><a NAME="E16:37_1_1_2"/>
<font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">FF</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="isocat_2.html#E6:37_1_1_2"><i><font color="Green" title="E37">A16</font></i></a>, <a class="txt" href="isocat_2.html#E13:37_1_1_2"><i><font color="Green" title="E41">A20</font></i></a>, <a class="txt" href="isocat_2.html#E14:37_1_1_2"><i><font color="Green" title="E42">A21</font></i></a>, <a class="ref" href="nattra_1.html#T42" title="NATTRA_1:th.42">NATTRA_1:42</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">FF</font> = <font color="Maroon" title="c4">FF</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of  <a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font>, <a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c5">FF</font>
;<br/>

<b>let </b><font color="Maroon" title="c6">F1</font>, <font color="Maroon" title="c7">F2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font>;<br/>

<b>assume </b><a NAME="E6:37_1_1"/><i><font color="Green" title="E35">A22</font></i>: 
<font color="Maroon" title="c6">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">F2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c8">t</font> be    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font>;<br/>

<a NAME="E7:37_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c8">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <span class="p1"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E6:37_1_1"><i><font color="Green" title="E35">A22</font></i></a>, <a class="ref" href="nattra_1.html#T35" title="NATTRA_1:th.35">NATTRA_1:35</a></i>;<br/>
<a NAME="E8:37_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c8">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2"><a href="cat_2.html#K11" title="CAT_2:func.11">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="cat_2.html#K11" title="CAT_2:func.11">:]</a></span>,<font color="Maroon" title="c3">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<b>hence </b><a NAME="E9:37_1_1"/>
<font color="Maroon" title="c5">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F1</font>,<font color="Maroon" title="c7">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c8">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c6">F1</font></span>)</span>,<span class="p3">(<span class="default"><a href="isocat_2.html#K5" title="ISOCAT_2:func.5">export</a> <font color="Maroon" title="c7">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="isocat_2.html#K6" title="ISOCAT_2:func.6">export</a> <font color="Maroon" title="c8">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="isocat_2.html#E3:37_1_1"><i><font color="Green" title="E34">A5</font></i></a></i>;<br/>


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