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

<span class="kw">let </span><font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c3">A2</font> be   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of  <a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K9" title="CATALG_1:func.9">homsym</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> ) &amp; ( 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  <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Maroon" title="c2">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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="Olive" title="b1">a</font> ) &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> holds <br/><span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span>)</span>,<font color="Maroon" title="c2">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b5">g</font>,<font color="Olive" title="b4">f</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b4">f</font> ) &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K9" title="CATALG_1:func.9">homsym</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> ) &amp; ( 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  <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Maroon" title="c3">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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="Olive" title="b1">a</font> ) &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> holds <br/><span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span>)</span>,<font color="Maroon" title="c3">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b5">g</font>,<font color="Olive" title="b4">f</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b4">f</font> ) implies <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:69_1_1"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K9" title="CATALG_1:func.9">homsym</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>
 <span class="kw">and </span><br/><a NAME="E2:69_1_1"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
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  <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Maroon" title="c2">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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="Olive" title="b1">a</font>
 <span class="kw">and </span><br/><a NAME="E3:69_1_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> holds <br/><span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span>)</span>,<font color="Maroon" title="c2">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b5">g</font>,<font color="Olive" title="b4">f</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b4">f</font>
 <span class="kw">and </span><br/><a NAME="E4:69_1_1"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K9" title="CATALG_1:func.9">homsym</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>
 <span class="kw">and </span><br/><a NAME="E5:69_1_1"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
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  <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Maroon" title="c3">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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="Olive" title="b1">a</font>
 <span class="kw">and </span><br/><a NAME="E6:69_1_1"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> holds <br/><span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span>)</span>,<font color="Maroon" title="c3">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b5">g</font>,<font color="Olive" title="b4">f</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b4">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A2</font></span><br/>

<div><span class="lab"><font color="Green" title="E34">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span> implies the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c2">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:69_1_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c2">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">o</font> = <font color="Maroon" title="c4">i</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:69_1_1_1_1"/>
( <font color="Maroon" title="c5">o</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">o</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="catalg_1.html#T25" target="_self" title="CATALG_1:th.25">Th25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_1_1_1"/>
<font color="Maroon" title="c5">o</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c2">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E3:69_1_1_1_1_1"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="catalg_1.html#T26" target="_self" title="CATALG_1:th.26">Th26</a></span>;<br/><a NAME="E4:69_1_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A9</font></span>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T55" title="CAT_1:th.55">CAT_1:55</a></span>;<br/><a NAME="E5:69_1_1_1_1_1"/><span class="lab"><font color="Green" title="E37">A10</font></span>: 
 <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalg_1.html#K9" title="CATALG_1:func.9">homsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="catalg_1.html#D5" target="_self" title="CATALG_1:def.5">Def5</a></span>;<br/><a NAME="E6:69_1_1_1_1_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K9" title="CATALG_1:func.9">homsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a></span>;<br/><a NAME="E7:69_1_1_1_1_1"/><span class="kw">then </span>
 <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="pralg_2.html#T10" title="PRALG_2:th.10">PRALG_2:10</a></span>;<br/><a NAME="E8:69_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A11</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c2">A1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c2">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><div><span class="lab"><font color="Green" title="E39">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c2">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c3">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:69_1_1_1_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c2">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c3">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><a NAME="E2:69_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A13</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:69_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c2">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</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="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69_1_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:69_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c2">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c3">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69_1_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E2:69_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E10:69_1_1_1_1_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K9" title="CATALG_1:func.9">homsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_1_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a></span>;<br/><a NAME="E11:69_1_1_1_1_1"/><span class="kw">then </span>
 <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c3">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="pralg_2.html#T10" title="PRALG_2:th.10">PRALG_2:10</a></span>;<br/><a NAME="E12:69_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A14</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c3">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c3">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E13:69_1_1_1_1_1"/>
(  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K8" title="CATALG_1:func.8">idsym</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c3">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="catalg_1.html#T34" target="_self" title="CATALG_1:th.34">Th34</a></span>;<br/><span class="kw">hence </span><a NAME="E14:69_1_1_1_1_1"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c2">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1_1_1_1"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E8:69_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E12:69_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="txt" href="#E9:69_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_1_1_2"/>
<font color="Maroon" title="c5">o</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c2">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E3:69_1_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A15</font></span>: 
<font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="catalg_1.html#T27" target="_self" title="CATALG_1:th.27">Th27</a></span>;<br/><div><span class="lab"><font color="Green" title="E36">A16</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:69_1_1_1_1_2_1"/>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</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> </span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">X</font> =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">X</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">g</font>, <font color="Maroon" title="c12">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:69_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E37">A17</font></span>: 
<font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>
 <span class="kw">and </span><br/><a NAME="E6:69_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E38">A18</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">and </span><br/><a NAME="E7:69_1_1_1_1_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c12">f</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#T2" title="FUNCT_6:th.2">FUNCT_6:2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">g</font> = <font color="Maroon" title="c11">g</font>, <font color="Maroon" title="c14">f</font> = <font color="Maroon" title="c12">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E37">A17</font></span></a>, <a class="txt" href="#E6:69_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E38">A18</font></span></a></span>;<br/><a NAME="E9:69_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E39">A19</font></span>: 
(  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E37">A17</font></span></a>, <a class="txt" href="#E6:69_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/><a NAME="E10:69_1_1_1_1_2_1"/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E37">A17</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/><a NAME="E11:69_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A20</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:69_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E39">A19</font></span></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></span>;<br/><a NAME="E12:69_1_1_1_1_2_1"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/><a NAME="E13:69_1_1_1_1_2_1"/><span class="kw">then </span>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c14">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:69_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E39">A19</font></span></a>, <a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></span>;<br/><span class="kw">hence </span><a NAME="E14:69_1_1_1_1_2_1"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:69_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E37">A21</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> implies <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<font color="Maroon" title="c2">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<font color="Maroon" title="c3">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:69_1_1_1_1_2_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<font color="Maroon" title="c2">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<font color="Maroon" title="c3">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">g</font>, <font color="Maroon" title="c11">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:69_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E38">A22</font></span>: 
<font color="Maroon" title="c10">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>
 <span class="kw">and </span><br/><a NAME="E4:69_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E39">A23</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
 <span class="kw">and </span><br/><a NAME="E5:69_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E40">A24</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c11">f</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#T2" title="FUNCT_6:th.2">FUNCT_6:2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c10">g</font>, <font color="Maroon" title="c13">f</font> = <font color="Maroon" title="c11">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="txt" href="#E4:69_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E39">A23</font></span></a></span>;<br/><a NAME="E7:69_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E41">A25</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c12">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c12">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/><a NAME="E8:69_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E42">A26</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/><a NAME="E9:69_1_1_1_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<font color="Maroon" title="c2">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c13">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E5:69_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="txt" href="#E7:69_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E41">A25</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:69_1_1_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<font color="Maroon" title="c2">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<font color="Maroon" title="c3">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69_1_1"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E5:69_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="txt" href="#E8:69_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="txt" href="#E7:69_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E41">A25</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:69_1_1_1_1_2"/><span class="lab"><font color="Green" title="E38">A27</font></span>: 
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="catalg_1.html#E51"><span class="lab"><font color="Green" title="E27">Lm3</font></span></a></span>;<br/><a NAME="E7:69_1_1_1_1_2"/><span class="lab"><font color="Green" title="E39">A28</font></span>: 
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<font color="Maroon" title="c3">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_1_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="catalg_1.html#E51"><span class="lab"><font color="Green" title="E27">Lm3</font></span></a></span>;<br/><a NAME="E8:69_1_1_1_1_2"/>
 <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<font color="Maroon" title="c3">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_1_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="catalg_1.html#E51"><span class="lab"><font color="Green" title="E27">Lm3</font></span></a></span>;<br/><a NAME="E9:69_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A29</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<font color="Maroon" title="c3">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<font color="Maroon" title="c3">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:69_1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A28</font></span></a>, <a class="txt" href="#E4:69_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E10:69_1_1_1_1_2"/>
 <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="catalg_1.html#E51"><span class="lab"><font color="Green" title="E27">Lm3</font></span></a></span>;<br/><a NAME="E11:69_1_1_1_1_2"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<font color="Maroon" title="c2">A1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K10" title="CATALG_1:func.10">compsym</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span>)</span>,<font color="Maroon" title="c2">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69_1_1_1_1_2"><span class="lab"><font color="Green" title="E38">A27</font></span></a>, <a class="txt" href="#E4:69_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E12:69_1_1_1_1_2"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c2">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1_1_1_2"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="txt" href="#E6:69_1_1_1_1_2"><span class="lab"><font color="Green" title="E38">A27</font></span></a>, <a class="txt" href="#E7:69_1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A28</font></span></a>, <a class="txt" href="#E9:69_1_1_1_1_2"><span class="lab"><font color="Green" title="E40">A29</font></span></a>, <a class="txt" href="#E5:69_1_1_1_1_2"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span> implies the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:69_1_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" title="CATALG_1:func.3">CatSign</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E3:69_1_1_2"/><span class="lab"><font color="Green" title="E35">A30</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalg_1.html#K9" title="CATALG_1:func.9">homsym</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="catalg_1.html#T24" target="_self" title="CATALG_1:th.24">Th24</a></span>;<br/><span class="kw">thus </span>the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> = 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E3:69_1_1_2"><span class="lab"><font color="Green" title="E35">A30</font></span></a></span>
<br/>.= 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_1_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E3:69_1_1_2"><span class="lab"><font color="Green" title="E35">A30</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E9:69_1_1"/><span class="kw">then </span>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:69_1_1"/>
<font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:69_1_1"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
