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<b>let </b><font color="Maroon">C</font> be   <a href="cat_4.html#NM1">Cartesian_category</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/>



<b>thus </b><a NAME="E1:42"/>
<font color="Maroon">a</font>,<font color="Maroon">a</font> <a href="cat_4.html#K2" target="_self">[x]</a> <span class="p1">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">C</font></span>)</span> <a href="cat_1.html#R1">are_isomorphic</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:42_1"/>
(  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="cat_4.html#K2" target="_self">[x]</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="cat_4.html#K2" target="_self">[x]</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">C</font></span>)</span></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_4.html#D2" target="_self">CAT_4:def 2</a><br/>

<b>take </b>
 <a href="cat_4.html#K13" target="_self">rho'</a> <font color="Maroon">a</font>
;<br/>

<b>take </b>
 <a href="cat_4.html#K12" target="_self">rho</a> <font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E2:42_1"/>
( <span class="p1">(<span class="default"><a href="cat_4.html#K13" target="_self">rho'</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K12" target="_self">rho</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="cat_4.html#K2" target="_self">[x]</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">C</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K12" target="_self">rho</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K13" target="_self">rho'</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:42"/>
(  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">C</font></span>)</span> <a href="cat_4.html#K2" target="_self">[x]</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">C</font></span>)</span> <a href="cat_4.html#K2" target="_self">[x]</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_4.html#D2" target="_self">CAT_4:def 2</a><br/>

<b>take </b>
 <a href="cat_4.html#K11" target="_self">lambda'</a> <font color="Maroon">a</font>
;<br/>

<b>take </b>
 <a href="cat_4.html#K10" target="_self">lambda</a> <font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E3:42"/>
( <span class="p1">(<span class="default"><a href="cat_4.html#K11" target="_self">lambda'</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K10" target="_self">lambda</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K1" target="_self">[1]</a> <font color="Maroon">C</font></span>)</span> <a href="cat_4.html#K2" target="_self">[x]</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K10" target="_self">lambda</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K11" target="_self">lambda'</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/>


</div>
