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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>













<b>assume </b><a NAME="E1:53"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>8</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D11">NATTRA_1:def 11</a><br/>

<b>given </b><font color="Maroon">c<sub>9</sub></font> being    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font><b> such that </b><a NAME="E2:53"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="nattra_1.html#V1">invertible</a>
 ;<br/>

<b>thus </b><a NAME="E3:53"/>
<font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="nattra_1.html#NR4" target="_self">~=</a> <font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>8</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:53_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="isocat_1.html#E1:53"><i><font color="Green">E38</font></i></a>, <a class="ref" href="isocat_1.html#T27" target="_self">Th27</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D11">NATTRA_1:def 11</a><br/>

<b>take </b>
<font color="Maroon">c<sub>6</sub></font> <a href="isocat_1.html#K4" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font>
;<br/>

<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>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D10">NATTRA_1:def 10</a><br/>

<a NAME="E2:53_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E1:53"><i><font color="Green">E38</font></i></a>, <a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/>
<a NAME="E3:53_1"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/>
<i><font color="Green">E41</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="isocat_1.html#K4" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="nattra_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E1:53"><i><font color="Green">E38</font></i></a>, <a class="ref" href="isocat_1.html#T28" target="_self">Th28</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E3:53_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
;<br/>
<a NAME="E5:53_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>10</sub></font> is <a href="cat_1.html#V5">invertible</a>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E2:53"><i><font color="Green">E39</font></i></a>, <a class="ref" href="nattra_1.html#D10">NATTRA_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E6:53_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="isocat_1.html#K4" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>10</sub></font> is <a href="cat_1.html#V5">invertible</a>
 <b>by </b><i><a class="txt" href="isocat_1.html#E4:53_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="isocat_1.html#T3" target="_self">Th3</a></i>;<br/>


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

<b>thus </b><a NAME="E4:53"/>
<font color="Maroon">c<sub>7</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>8</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="isocat_1.html#E1:53"><i><font color="Green">E38</font></i></a>, <a class="ref" href="isocat_1.html#T27" target="_self">Th27</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D11">NATTRA_1:def 11</a><br/>

<b>take </b>
<font color="Maroon">c<sub>9</sub></font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>
;<br/>

<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>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D10">NATTRA_1:def 10</a><br/>

<a NAME="E5:53"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <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>9</sub></font> <a href="nattra_1.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E1:53"><i><font color="Green">E38</font></i></a>, <a class="ref" href="isocat_1.html#T29" target="_self">Th29</a></i>;<br/>
<b>hence </b><a NAME="E6:53"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c<sub>10</sub></font> is <a href="cat_1.html#V5">invertible</a>
 <b>by </b><i><a class="txt" href="isocat_1.html#E2:53"><i><font color="Green">E39</font></i></a>, <a class="ref" href="nattra_1.html#D10">NATTRA_1:def 10</a></i>;<br/>


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