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

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_4.html#NM1" title="CAT_4:NM.1">Cartesian_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#K2" title="CAT_4:func.2">[x]</a> <span class="p1">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</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#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</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#K2" title="CAT_4:func.2">[x]</a> <span class="p1">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</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#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c2">a</font> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a>  )</span><br/>



<b>thus </b><a NAME="E1:43"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p1">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</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#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</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="43_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:43_1"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">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#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</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>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</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="ref" href="cat_4.html#T21" target="_self" title="CAT_4:th.21">Th21</a>, <a class="ref" href="cat_4.html#T30" target="_self" title="CAT_4:th.30">Th30</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">  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#K2" title="CAT_4:func.2">[x]</a> <span class="p1">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</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#K2" title="CAT_4:func.2">[x]</a> <span class="p1">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</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#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b2">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</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#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<b>take </b>
 <a href="cat_4.html#K13" title="CAT_4:func.13">rho'</a> <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 <font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p1">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c2">a</font> st <br/>( <span class="p1">(<span class="default"><a href="cat_4.html#K13" title="CAT_4:func.13">rho'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</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#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K13" title="CAT_4:func.13">rho'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<b>take </b>
 <a href="cat_4.html#K12" title="CAT_4:func.12">rho</a> <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"> ( <span class="p1">(<span class="default"><a href="cat_4.html#K13" title="CAT_4:func.13">rho'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K12" title="CAT_4:func.12">rho</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K12" title="CAT_4:func.12">rho</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K13" title="CAT_4:func.13">rho'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<b>thus </b><a NAME="E2:43_1"/>
( <span class="p1">(<span class="default"><a href="cat_4.html#K13" title="CAT_4:func.13">rho'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K12" title="CAT_4:func.12">rho</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K12" title="CAT_4:func.12">rho</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K13" title="CAT_4:func.13">rho'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="ref" href="cat_4.html#T31" target="_self" title="CAT_4:th.31">Th31</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="E2:43"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">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#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</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>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</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="ref" href="cat_4.html#T21" target="_self" title="CAT_4:th.21">Th21</a>, <a class="ref" href="cat_4.html#T30" target="_self" title="CAT_4:th.30">Th30</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">  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#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</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#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</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#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b2">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp; <font color="Olive" title="b2">f'</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<b>take </b>
 <a href="cat_4.html#K11" title="CAT_4:func.11">lambda'</a> <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#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> st <br/>( <span class="p1">(<span class="default"><a href="cat_4.html#K11" title="CAT_4:func.11">lambda'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp; <font color="Olive" title="b1">f'</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K11" title="CAT_4:func.11">lambda'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<b>take </b>
 <a href="cat_4.html#K10" title="CAT_4:func.10">lambda</a> <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"> ( <span class="p1">(<span class="default"><a href="cat_4.html#K11" title="CAT_4:func.11">lambda'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K10" title="CAT_4:func.10">lambda</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K10" title="CAT_4:func.10">lambda</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K11" title="CAT_4:func.11">lambda'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<b>thus </b><a NAME="E3:43"/>
( <span class="p1">(<span class="default"><a href="cat_4.html#K11" title="CAT_4:func.11">lambda'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K10" title="CAT_4:func.10">lambda</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_4.html#K1" title="CAT_4:func.1">[1]</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="cat_4.html#K2" title="CAT_4:func.2">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_4.html#K10" title="CAT_4:func.10">lambda</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_4.html#K11" title="CAT_4:func.11">lambda'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="ref" href="cat_4.html#T31" target="_self" title="CAT_4:th.31">Th31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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