<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="nattra_1.html#M2">natural_transformation</a> of  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">c<sub>4</sub></font>, <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:32_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>7</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<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="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:32_1_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>8</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<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="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<a NAME="E3:32_1_1"/>
 <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E1:32"><i><font color="Green">E28</font></i></a>, <a class="ref" href="isocat_2.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E4:32_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">c<sub>4</sub></font> <a href="nattra_1.html#R1">is_transformable_to</a>  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/>
<a NAME="E5:32_1_1"/>
the <a href="cat_1.html#U1">Objects</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="cat_2.html#T33">CAT_2:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:32_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>10</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="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="isocat_2.html#E1:32_1_1"><i><font color="Green">E29</font></i></a></i>;<br/><b>hence </b><a NAME="E2:32_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="isocat_2.html#E2:32_1_1"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:32_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="isocat_2.html#E4:32_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="nattra_1.html#T20">NATTRA_1:20</a></i>;<br/>


</div>
