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

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_4.html#NM2" title="CAT_4:NM.2">Cocartesian_category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a>  &amp; <font color="Olive" title="b1">a</font>,<span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Olive" title="b1">a</font> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a>  &amp; <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a>  )</span><br/>



<a NAME="E1:101"/><i><font color="Green" title="E48">A1</font></i>: 
(  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="cat_4.html#T60" target="_self" title="CAT_4:th.60">Th60</a>, <a class="ref" href="cat_1.html#T56" title="CAT_1:th.56">CAT_1:56</a></i>;<br/>
<b>thus </b><a NAME="E2:101"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:101_1"/><i><font color="Green" title="E49">A2</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="cat_4.html#T66" target="_self" title="CAT_4:th.66">Th66</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_4.html#D2" target="_self" title="CAT_4:def.2">CAT_4:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  ex <font color="Olive" title="b1">f</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> ex <font color="Olive" title="b2">f'</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b2">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">f'</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font> ) )</span><br/>

<b>thus </b><a NAME="E2:101_1"/>
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="cat_4.html#E1:101"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="cat_4.html#T70" target="_self" title="CAT_4:th.70">Th70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> ex <font color="Olive" title="b2">f'</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b2">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">f'</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<b>take </b><font color="Maroon" title="c3">g</font> =  <a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">f'</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font> st <br/>( <font color="Maroon" title="c3">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b1">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b1">f'</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<b>take </b><font color="Maroon" title="c4">f</font> = <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K24" title="CAT_4:func.24">init</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<a NAME="E3:101_1"/>
( <span class="p1">(<span class="default"><a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <font color="Maroon" title="c2">a</font>,<span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <font color="Maroon" title="c2">a</font>,<span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K24" title="CAT_4:func.24">init</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_4.html#K26" title="CAT_4:func.26">in2</a> <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> &amp;  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:101_1"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="cat_4.html#T59" target="_self" title="CAT_4:th.59">Th59</a>, <a class="ref" href="cat_4.html#T66" target="_self" title="CAT_4:th.66">Th66</a>, <a class="ref" href="cat_1.html#T58" title="CAT_1:th.58">CAT_1:58</a></i>;<br/>
<a NAME="E4:101_1"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K26" title="CAT_4:func.26">in2</a> <font color="Maroon" title="c2">a</font>,<span class="p3">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span>
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:101"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="cat_4.html#E1:101_1"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="cat_4.html#T72" target="_self" title="CAT_4:th.72">Th72</a></i>;<br/>
<b>hence </b><a NAME="E5:101_1"/>
( <font color="Maroon" title="c3">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="txt" href="cat_4.html#E1:101"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="cat_4.html#D29" target="_self" title="CAT_4:def.29">Def29</a>, <a class="ref" href="cat_4.html#T71" target="_self" title="CAT_4:th.71">Th71</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E3:101"/><i><font color="Green" title="E49">A3</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="cat_4.html#T66" target="_self" title="CAT_4:th.66">Th66</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_4.html#D2" target="_self" title="CAT_4:def.2">CAT_4:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  ex <font color="Olive" title="b1">f</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font> ex <font color="Olive" title="b2">f'</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b2">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp; <font color="Olive" title="b2">f'</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font> ) )</span><br/>

<b>thus </b><a NAME="E4:101"/>
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="cat_4.html#E1:101"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="cat_4.html#T70" target="_self" title="CAT_4:th.70">Th70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font> ex <font color="Olive" title="b2">f'</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b2">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp; <font color="Olive" title="b2">f'</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<b>take </b><font color="Maroon" title="c3">g</font> =  <a href="cat_4.html#K26" title="CAT_4:func.26">in2</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">f'</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> st <br/>( <font color="Maroon" title="c3">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b1">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp; <font color="Olive" title="b1">f'</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<b>take </b><font color="Maroon" title="c4">f</font> = <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K24" title="CAT_4:func.24">init</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp; <font color="Maroon" title="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<a NAME="E5:101"/>
( <span class="p1">(<span class="default"><a href="cat_4.html#K26" title="CAT_4:func.26">in2</a> <span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_4.html#K26" title="CAT_4:func.26">in2</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K26" title="CAT_4:func.26">in2</a> <span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K24" title="CAT_4:func.24">init</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font> &amp;  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="cat_4.html#E3:101"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="cat_4.html#T59" target="_self" title="CAT_4:th.59">Th59</a>, <a class="ref" href="cat_4.html#T66" target="_self" title="CAT_4:th.66">Th66</a>, <a class="ref" href="cat_1.html#T58" title="CAT_1:th.58">CAT_1:58</a></i>;<br/>
<a NAME="E6:101"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K25" title="CAT_4:func.25">in1</a> <span class="p3">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_4.html#K26" title="CAT_4:func.26">in2</a> <span class="p3">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font></span>)</span></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span>
 
<b>by </b><i><a class="txt" href="cat_4.html#E1:101"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="cat_4.html#E3:101"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="cat_4.html#T72" target="_self" title="CAT_4:th.72">Th72</a></i>;<br/>
<b>hence </b><a NAME="E7:101"/>
( <font color="Maroon" title="c3">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K19" title="CAT_4:func.19">[0]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp; <font color="Maroon" title="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="txt" href="cat_4.html#E1:101"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="cat_4.html#D29" target="_self" title="CAT_4:def.29">Def29</a>, <a class="ref" href="cat_4.html#T71" target="_self" title="CAT_4:th.71">Th71</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
