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

<b>let </b><font color="Maroon">A1</font> be  <a href="msualg_1.html#V4">strict</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of  <a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <font color="Maroon">A2</font> be  <a href="msualg_1.html#V4">strict</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of  <a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:70_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> holds  the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">a</font>,<font color="Olive">b</font>
 <b>and </b><br/><a NAME="E2:70_1_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">a</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> holds  <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Olive">a</font></span>)</span>,<font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Olive">a</font>
 <b>and </b><br/><a NAME="E3:70_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st  <a href="cat_1.html#K2">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> holds <br/><span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font></span>)</span>,<font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="cat_1.html#K4">*</a> <font color="Olive">f</font>
 <b>and </b><br/><a NAME="E4:70_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> holds  the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">a</font>,<font color="Olive">b</font>
 <b>and </b><br/><a NAME="E5:70_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">a</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> holds  <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Olive">a</font></span>)</span>,<font color="Maroon">A2</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Olive">a</font>
 <b>and </b><br/><a NAME="E6:70_1_1"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st  <a href="cat_1.html#K2">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> holds <br/><span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font></span>)</span>,<font color="Maroon">A2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="cat_1.html#K4">*</a> <font color="Olive">f</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:70_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Maroon">b</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:70_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i/>;<br/><b>thus </b>the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="catalg_1.html#E1:70_1_1"><i><font color="Green">E37</font></i></a>, </i>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<a NAME="E8:70_1_1"/><b>then </b><i><font color="Green">E76</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:70_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">o</font> = <font color="Maroon">i</font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font></span>)</span> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_1_2_1"/>
( <font color="Maroon">o</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">o</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:70_1_1_2_1_1"/>
<font color="Maroon">o</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:70_1_1_2_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><a NAME="E4:70_1_1_2_1_1"/>
(  <a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">a</font>,<font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">a</font>,<font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E1:70_1_1"><i><font color="Green">E37</font></i></a>, , </i>;<br/><a NAME="E5:70_1_1_2_1_1"/><b>then </b>
(  <a href="msualg_1.html#K4">Result</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> &amp;  <a href="msualg_1.html#K4">Result</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> &amp;  <a href="cat_1.html#K10">id</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T55">CAT_1:55</a>, <a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>;<br/><a NAME="E6:70_1_1_2_1_1"/><b>then </b><i><font color="Green">E78</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">A1</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">A2</font> &amp;  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> &amp;  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:70_1_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:70_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:70_1_1_2_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">A2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="txt" href="catalg_1.html#E2:70_1_1"><i><font color="Green">E41</font></i></a></i>;<br/><b>hence </b><a NAME="E4:70_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">A2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div><a NAME="E8:70_1_1_2_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
 <a href="msualg_1.html#K5">Den</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">A2</font>
 <b>by </b><i><a class="txt" href="catalg_1.html#E4:70_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>thus </b><a NAME="E9:70_1_1_2_1_1"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="catalg_1.html#E5:70_1_1"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:70_1_1_2_1_2"/>
<font color="Maroon">o</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Maroon">b</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Maroon">c</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:70_1_1_2_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 <b>by </b><i/>;<br/><a NAME="E4:70_1_1_2_1_2"/><i><font color="Green">E84</font></i>: 
(  <a href="msualg_1.html#K4">Result</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp;  <a href="msualg_1.html#K4">Result</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp;  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> &amp;  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E1:70_1_1"><i><font color="Green">E37</font></i></a>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_1_1_2_1_2_1"/>
 <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">X</font> =  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>consider </b><font color="Maroon">g</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">f</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:70_1_1_2_1_2_1"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="finseq_1.html#K10">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="funct_6.html#T2">FUNCT_6:2</a></i>;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font>, <font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> <b>by </b><i><a class="txt" href="catalg_1.html#E5:70_1_1"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E7:70_1_1_2_1_2_1"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E5:70_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/><a NAME="E8:70_1_1_2_1_2_1"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/><b>hence </b><a NAME="E9:70_1_1_2_1_2_1"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">c</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_1.html#T18">CAT_1:18</a></i>;<br/></div><b>end;</b></div><a NAME="E6:70_1_1_2_1_2"/><b>then </b><i><font color="Green">E87</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<font color="Maroon">A1</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<font color="Maroon">A2</font> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E4:70_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:70_1_1_2_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 ;<br/><b>then consider </b><font color="Maroon">g</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">f</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:70_1_1_2_1_2_2"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="finseq_1.html#K10">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="funct_6.html#T2">FUNCT_6:2</a></i>;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font>, <font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> <b>by </b><i><a class="ref" href="catalg_1.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E5:70_1_1_2_1_2_2"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="catalg_1.html#T4" target="_self">Th4</a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/><a NAME="E6:70_1_1_2_1_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<font color="Maroon">A2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="txt" href="catalg_1.html#E3:70_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="catalg_1.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E7:70_1_1_2_1_2_2"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<font color="Maroon">A2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div><a NAME="E8:70_1_1_2_1_2"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="msualg_1.html#K5">Den</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<font color="Maroon">A2</font>
 <b>by </b><i><a class="txt" href="catalg_1.html#E4:70_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="catalg_1.html#E5:70_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>thus </b><a NAME="E9:70_1_1_2_1_2"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="catalg_1.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:70_1_1"/>
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
