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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>assume </b><a NAME="E1:62"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="isocat_1.html#NR5" target="_self">are_equivalent</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="isocat_1.html#M1" target="_self">Equivalence</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:62"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="nattra_1.html#NR4" target="_self">~=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="nattra_1.html#NR4" target="_self">~=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="isocat_1.html#E1:62"><i><font color="Green">E43</font></i></a>, <a class="ref" href="isocat_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E4:62"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="isocat_1.html#M1" target="_self">Equivalence</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:62_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="isocat_1.html#NR5" target="_self">are_equivalent</a> 
 <b>by </b><i><a class="txt" href="isocat_1.html#E1:62"><i><font color="Green">E43</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="isocat_1.html#D11" target="_self">ISOCAT_1:def 11</a><br/>

<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E2:62_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="nattra_1.html#NR4" target="_self">~=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="nattra_1.html#NR4" target="_self">~=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="isocat_1.html#E3:62"><i><font color="Green">E44</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:62"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="isocat_1.html#M1" target="_self">Equivalence</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="nattra_1.html#NR4" target="_self">~=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K14">*</a> <font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#NR4" target="_self">~=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="isocat_1.html#E3:62"><i><font color="Green">E44</font></i></a></i>;<br/>


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