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<b>let </b><font color="Maroon">A</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">F1</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">F2</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">F3</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>









<b>assume </b><a NAME="E1:52"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">F1</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D11" target="_self">NATTRA_1:def 11</a><br/>

<b>given </b><font color="Maroon">t</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><a NAME="E2:52"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">t</font> is <a href="nattra_1.html#V1" target="_self">invertible</a>
 ;<br/>

<b>assume </b><a NAME="E3:52"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">F2</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F3</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D11" target="_self">NATTRA_1:def 11</a><br/>

<b>given </b><font color="Maroon">t'</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F2</font>,<font color="Maroon">F3</font><b> such that </b><a NAME="E4:52"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">t'</font> is <a href="nattra_1.html#V1" target="_self">invertible</a>
 ;<br/>

<b>thus </b><a NAME="E5:52"/>
<font color="Maroon">F1</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F3</font>
 <b>by </b><i>, , </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D11" target="_self">NATTRA_1:def 11</a><br/>

<b>take </b>
<font color="Maroon">t'</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t</font>
;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D10" target="_self">NATTRA_1:def 10</a><br/>

<a NAME="E6:52"/><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">t'</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t</font></span>)</span> <a href="nattra_1.html#K5" target="_self">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t'</font> <a href="nattra_1.html#K5" target="_self">.</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5" target="_self">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:52"/>
( <font color="Maroon">F1</font> <a href="nattra_1.html#R1" target="_self">is_transformable_to</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">F2</font> <a href="nattra_1.html#R1" target="_self">is_transformable_to</a> <font color="Maroon">F3</font> )
 
<b>by </b><i>, , <a class="txt" href="nattra_1.html#E4:52"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E8:52"/><b>then </b><i><font color="Green">E33</font></i>: 
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F3</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</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#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E9:52"/>
( <font color="Maroon">t'</font> <a href="nattra_1.html#K5" target="_self">.</a> <font color="Maroon">a</font> is <a href="cat_1.html#V5">invertible</a> &amp; <font color="Maroon">t</font> <a href="nattra_1.html#K5" target="_self">.</a> <font color="Maroon">a</font> is <a href="cat_1.html#V5">invertible</a> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E10:52"/>
<span class="p1">(<span class="default"><font color="Maroon">t'</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t</font></span>)</span> <a href="nattra_1.html#K5" target="_self">.</a> <font color="Maroon">a</font> is <a href="cat_1.html#V5">invertible</a>
 <b>by </b><i>, , <a class="ref" href="cat_1.html#T75">CAT_1:75</a></i>;<br/>


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