<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_4.html#NM2">Cocartesian_category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:97"/><i><font color="Green">E56</font></i>: 
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <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> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="cat_4.html#T60" target="_self">Th60</a>, <a class="ref" href="cat_1.html#T56">CAT_1:56</a></i>;<br/>
<b>thus </b><a NAME="E2:97"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="cat_4.html#K20" target="_self">+</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cat_1.html#R1">are_isomorphic</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:97_1"/><i><font color="Green">E57</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_4.html#K20" target="_self">+</a> <span class="p2">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="cat_4.html#T66" target="_self">Th66</a></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>thus </b><a NAME="E2:97_1"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_4.html#K20" target="_self">+</a> <span class="p2">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="cat_4.html#E1:97"><i><font color="Green">E56</font></i></a>, <a class="ref" href="cat_4.html#T70" target="_self">Th70</a></i>;<br/>

<b>take </b><font color="Maroon">c<sub>3</sub></font> =  <a href="cat_4.html#K25" target="_self">in1</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>take </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="cat_4.html#K27" target="_self">[$</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K24" target="_self">init</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_4.html#K27" target="_self">$]</a></span>;<br/>

<a NAME="E3:97_1"/>
( <span class="p1">(<span class="default"><a href="cat_4.html#K25" target="_self">in1</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_4.html#K25" target="_self">in1</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K25" target="_self">in1</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K24" target="_self">init</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_4.html#K26" target="_self">in2</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_4.html#K20" target="_self">+</a> <span class="p2">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:97_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="cat_4.html#T59" target="_self">Th59</a>, <a class="ref" href="cat_4.html#T66" target="_self">Th66</a>, <a class="ref" href="cat_1.html#T58">CAT_1:58</a></i>;<br/>
<a NAME="E4:97_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_4.html#K27" target="_self">[$</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K25" target="_self">in1</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K26" target="_self">in2</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="cat_4.html#K27" target="_self">$]</a></span>
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:97"><i><font color="Green">E56</font></i></a>, <a class="txt" href="cat_4.html#E1:97_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="cat_4.html#T72" target="_self">Th72</a></i>;<br/>
<b>hence </b><a NAME="E5:97_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_4.html#K20" target="_self">+</a> <span class="p2">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="cat_4.html#E1:97"><i><font color="Green">E56</font></i></a>, <a class="ref" href="cat_4.html#D29" target="_self">Def29</a>, <a class="ref" href="cat_4.html#T71" target="_self">Th71</a></i>;<br/>


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

<b>thus </b><a NAME="E3:97"/><i><font color="Green">E57</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cat_4.html#K20" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="cat_4.html#T66" target="_self">Th66</a></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>thus </b><a NAME="E4:97"/>
 <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#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cat_4.html#K20" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="cat_4.html#E1:97"><i><font color="Green">E56</font></i></a>, <a class="ref" href="cat_4.html#T70" target="_self">Th70</a></i>;<br/>

<b>take </b><font color="Maroon">c<sub>3</sub></font> =  <a href="cat_4.html#K26" target="_self">in2</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>;<br/>

<b>take </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="cat_4.html#K27" target="_self">[$</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K24" target="_self">init</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_4.html#K27" target="_self">$]</a></span>;<br/>

<a NAME="E5:97"/>
( <span class="p1">(<span class="default"><a href="cat_4.html#K26" target="_self">in2</a> <span class="p2">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_4.html#K26" target="_self">in2</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K26" target="_self">in2</a> <span class="p2">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K24" target="_self">init</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_4.html#K25" target="_self">in1</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cat_4.html#K20" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="cat_4.html#E3:97"><i><font color="Green">E57</font></i></a>, <a class="ref" href="cat_4.html#T59" target="_self">Th59</a>, <a class="ref" href="cat_4.html#T66" target="_self">Th66</a>, <a class="ref" href="cat_1.html#T58">CAT_1:58</a></i>;<br/>
<a NAME="E6:97"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_4.html#K27" target="_self">[$</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K25" target="_self">in1</a> <span class="p3">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K26" target="_self">in2</a> <span class="p3">(<span class="default"><a href="cat_4.html#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_4.html#K27" target="_self">$]</a></span>
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:97"><i><font color="Green">E56</font></i></a>, <a class="txt" href="cat_4.html#E3:97"><i><font color="Green">E57</font></i></a>, <a class="ref" href="cat_4.html#T72" target="_self">Th72</a></i>;<br/>
<b>hence </b><a NAME="E7:97"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font> <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#K19" target="_self">[0]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cat_4.html#K20" target="_self">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="cat_4.html#E1:97"><i><font color="Green">E56</font></i></a>, <a class="ref" href="cat_4.html#D29" target="_self">Def29</a>, <a class="ref" href="cat_4.html#T71" target="_self">Th71</a></i>;<br/>


</div>
