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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/>



<b>assume </b><a NAME="E1:62"/><i><font color="Green" title="E41">A1</font></i>: 
<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="isocat_1.html#NR5" title="ISOCAT_1:NR.5">are_equivalent</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c3">F</font> be    <a href="isocat_1.html#M1" title="ISOCAT_1:mode.1">Equivalence</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>;<br/>

<b>consider </b><font color="Maroon" title="c4">G</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:62"/><i><font color="Green" title="E42">A2</font></i>: 
( <font color="Maroon" title="c4">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">F</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">G</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c2">B</font> )
 <b>by </b><i><a class="txt" href="isocat_1.html#E1:62"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="isocat_1.html#D11" target="_self" title="ISOCAT_1:def.11">Def11</a></i>;<br/>
<a NAME="E4:62"/>
<font color="Maroon" title="c4">G</font> is    <a href="isocat_1.html#M1" title="ISOCAT_1:mode.1">Equivalence</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</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" title="c2">B</font>,<font color="Maroon" title="c1">A</font> <a href="isocat_1.html#NR5" title="ISOCAT_1:NR.5">are_equivalent</a> 
 <b>by </b><i><a class="txt" href="isocat_1.html#E1:62"><i><font color="Green" title="E41">A1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="isocat_1.html#D11" target="_self" title="ISOCAT_1:def.11">ISOCAT_1:def 11</a><br/>

<b>take </b>
<font color="Maroon" title="c3">F</font>
;<br/>

<b>thus </b><a NAME="E2:62_1"/>
( <font color="Maroon" title="c3">F</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">G</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c4">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="txt" href="isocat_1.html#E3:62"><i><font color="Green" title="E42">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:62"/>
 ex <font color="Olive" title="b1">G</font> being   <a href="isocat_1.html#M1" title="ISOCAT_1:mode.1">Equivalence</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c3">F</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">F</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Olive" title="b1">G</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c2">B</font> )
 <b>by </b><i><a class="txt" href="isocat_1.html#E3:62"><i><font color="Green" title="E42">A2</font></i></a></i>;<br/>


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