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





<b>assume </b><a NAME="E1:61"/><i><font color="Green">E42</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>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="isocat_1.html#NR5" target="_self">are_equivalent</a>  )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</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>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="isocat_1.html#M1" target="_self">Equivalence</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>6</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="E3:61"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>6</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>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>6</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#E1:61"><i><font color="Green">E42</font></i></a>, <a class="ref" href="isocat_1.html#D11" target="_self">Def11</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:61"/><i><font color="Green">E44</font></i>: 
( <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#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>5</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>  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="isocat_1.html#E1:61"><i><font color="Green">E42</font></i></a>, <a class="ref" href="isocat_1.html#D11" target="_self">Def11</a></i>;<br/>
<b>thus </b><a NAME="E6:61"/>
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="isocat_1.html#NR5" target="_self">are_equivalent</a> 
 <b>by </b><i><a class="txt" href="isocat_1.html#E1:61"><i><font color="Green">E42</font></i></a>, <a class="ref" href="isocat_1.html#T53" target="_self">Th53</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>6</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>7</sub></font>
;<br/>

<a NAME="E7:61"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="isocat_1.html#T4" target="_self">Th4</a>, <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">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="nattra_1.html#NR4" target="_self">~=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E5:61"><i><font color="Green">E44</font></i></a>, <a class="ref" href="isocat_1.html#T49" target="_self">Th49</a></i>;<br/>
<a NAME="E9:61"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>4</sub></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">c<sub>6</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>4</sub></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">c<sub>6</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <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>4</sub></font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E8:61"><i><font color="Green">E45</font></i></a>, <a class="ref" href="isocat_1.html#T48" target="_self">Th48</a></i>;<br/>
<b>hence </b><a NAME="E11:61"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <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#E3:61"><i><font color="Green">E43</font></i></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">c<sub>5</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="isocat_1.html#T4" target="_self">Th4</a>, <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">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="nattra_1.html#NR4" target="_self">~=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E3:61"><i><font color="Green">E43</font></i></a>, <a class="ref" href="isocat_1.html#T49" target="_self">Th49</a></i>;<br/>
<a NAME="E14:61"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>7</sub></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">c<sub>5</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>7</sub></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">c<sub>5</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nattra_1.html#NR4" target="_self">~=</a> <font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="isocat_1.html#E13:61"><i><font color="Green">E46</font></i></a>, <a class="ref" href="isocat_1.html#T48" target="_self">Th48</a></i>;<br/>
<b>hence </b><a NAME="E16:61"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nattra_1.html#NR4" target="_self">~=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="isocat_1.html#E5:61"><i><font color="Green">E44</font></i></a>, <a class="ref" href="nattra_1.html#T32">NATTRA_1:32</a></i>;<br/>


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