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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</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>assume </b><a NAME="E1:51"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font><b> such that </b><a NAME="E2:51"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="nattra_1.html#V1" target="_self">invertible</a>
 ;<br/>

<b>thus </b><a NAME="E3:51"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E1:51"><i><font color="Green">E42</font></i></a>, <a class="txt" href="nattra_1.html#E2:51"><i><font color="Green">E43</font></i></a>, <a class="txt" href="nattra_1.html#E45"><i><font color="Green">Lemma39</font></i></a></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">c<sub>5</sub></font> <a href="nattra_1.html#K10" target="_self">"</a> 
;<br/>

<b>let </b><font color="Maroon">c<sub>6</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" target="_self">NATTRA_1:def 10</a><br/>

<a NAME="E4:51"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nattra_1.html#R1" target="_self">is_transformable_to</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E1:51"><i><font color="Green">E42</font></i></a>, <a class="ref" href="nattra_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E5:51"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>6</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" target="_self">Def2</a></i>;<br/>
<a NAME="E6:51"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="nattra_1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> is <a href="cat_1.html#V5">invertible</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nattra_1.html#K10" target="_self">"</a> </span>)</span> <a href="nattra_1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nattra_1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="cat_1.html#K11">"</a>  )
 
<b>by </b><i><a class="txt" href="nattra_1.html#E1:51"><i><font color="Green">E42</font></i></a>, <a class="txt" href="nattra_1.html#E2:51"><i><font color="Green">E43</font></i></a>, <a class="ref" href="nattra_1.html#D10" target="_self">Def10</a>, <a class="ref" href="nattra_1.html#T30" target="_self">Th30</a></i>;<br/>
<b>hence </b><a NAME="E7:51"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nattra_1.html#K10" target="_self">"</a> </span>)</span> <a href="nattra_1.html#K5" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> is <a href="cat_1.html#V5">invertible</a>
 <b>by </b><i><a class="txt" href="nattra_1.html#E5:51"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cat_1.html#T76">CAT_1:76</a></i>;<br/>


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