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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">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/>





<b>assume </b><a NAME="E1:61"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="isocat_1.html#NR5" target="_self">are_equivalent</a>  &amp; <font color="Maroon">B</font>,<font color="Maroon">C</font> <a href="isocat_1.html#NR5" target="_self">are_equivalent</a>  )
 ;<br/>

<b>let </b><font color="Maroon">F</font> be    <a href="isocat_1.html#M1" target="_self">Equivalence</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="isocat_1.html#M1" target="_self">Equivalence</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/>



<b>consider </b><font color="Maroon">F'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:61"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">F'</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font> <a href="nattra_1.html#NR4" target="_self">~=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">A</font> &amp; <font color="Maroon">F</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F'</font> <a href="nattra_1.html#NR4" target="_self">~=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">B</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">G'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">C</font>,<font color="Maroon">B</font><b> such that </b><br/><a NAME="E5:61"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">G'</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G</font> <a href="nattra_1.html#NR4" target="_self">~=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">B</font> &amp; <font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G'</font> <a href="nattra_1.html#NR4" target="_self">~=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">C</font> )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E6:61"/>
<font color="Maroon">A</font>,<font color="Maroon">C</font> <a href="isocat_1.html#NR5" target="_self">are_equivalent</a> 
 <b>by </b><i>, <a class="ref" href="isocat_1.html#T8" target="_self">Th8</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">F'</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G'</font>
;<br/>

<a NAME="E7:61"/>
( <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G'</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">G</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F'</font> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">F'</font> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">B</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">F'</font> )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>
<a NAME="E8:61"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G'</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">G</font> <a href="nattra_1.html#NR4" target="_self">~=</a> <font color="Maroon">F'</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:61"/>
<span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G'</font></span>)</span> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F'</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G'</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">G</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>
<a NAME="E10:61"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G'</font></span>)</span> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#NR4" target="_self">~=</a> <font color="Maroon">F'</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E11:61"/>
<span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G'</font></span>)</span> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#NR4" target="_self">~=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="isocat_1.html#T9" target="_self">Th9</a>, <a class="ref" href="nattra_1.html#T32">NATTRA_1:32</a></i>;<br/>

<a NAME="E12:61"/>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">F'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F'</font></span>)</span> &amp; <font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">B</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>
<a NAME="E13:61"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">F'</font> <a href="nattra_1.html#NR4" target="_self">~=</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="isocat_1.html#T9" target="_self">Th9</a>, </i>;<br/>
<a NAME="E14:61"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G'</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">F'</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">G'</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>
<a NAME="E15:61"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G'</font></span>)</span> <a href="nattra_1.html#NR4" target="_self">~=</a> <font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G'</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E16:61"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G'</font></span>)</span> <a href="nattra_1.html#NR4" target="_self">~=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="nattra_1.html#T32">NATTRA_1:32</a></i>;<br/>


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