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

<span class="kw">reconsider </span><font color="Maroon" title="c9">O</font> =  <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font>, <font color="Maroon" title="c10">M</font> =  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> 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">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">$1,$1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p4">(<span class="default">$1 <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p4">(<span class="default">$1 <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> = $1 <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> = $1 <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">D9</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:10_1_1"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> holds  <font color="Maroon" title="c11">D9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">k</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span>
<a NAME="E4:10_1_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
 <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">o1</font>,<font color="Olive" title="b4">o2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">h</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">g</font> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>, <font color="Olive" title="b2">h</font> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">D</font>, <font color="Olive" title="b3">o1</font>, <font color="Olive" title="b4">o2</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> : (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">o1</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">o2</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a>  &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">o1</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">o2</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a>  &amp; <span class="p1">(<span class="default"><font color="Olive" title="b4">o2</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">h</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">o1</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="commacat.html#D6" target="_self" title="COMMACAT:def.6">Def6</a></span>;<br/>
<a NAME="E5:10_1_1"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
for <font color="Olive" title="b1">o</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font>  ex <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">o</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">o</font>,<font color="Olive" title="b1">k</font>]</span><br/>

<a NAME="E1:10_1_1_1"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>;<br/>
<a NAME="E2:10_1_1_1"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="commacat.html#T2" target="_self" title="COMMACAT:th.2">Th2</a>, <a class="ref" href="cat_1.html#T108" title="CAT_1:th.108">CAT_1:108</a></span>;<br/>
<a NAME="E3:10_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T47" title="CAT_1:th.47">CAT_1:47</a></span>;<br/>
<a NAME="E4:10_1_1_1"/>
(  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span> &amp; <font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="commacat.html#T2" target="_self" title="COMMACAT:th.2">Th2</a>, <a class="ref" href="cat_1.html#T108" title="CAT_1:th.108">CAT_1:108</a></span>;<br/>
<a NAME="E5:10_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T46" title="CAT_1:th.46">CAT_1:46</a></span>;<br/>
<a NAME="E6:10_1_1_1"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>;<br/>
<a NAME="E7:10_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c12">o</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_1_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E1:10_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E3:10_1_1_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="txt" href="#E5:10_1_1_1"><span class="lab"><font color="Green" title="E13">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:10_1_1_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">o</font>,<font color="Olive" title="b1">k</font>]
 ; <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>
<span class="kw">consider </span><font color="Maroon" title="c12">I</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:10_1_1"/><span class="lab"><font color="Green" title="E11">A8</font></span>: 
for <font color="Olive" title="b1">o</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c12">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E5:10_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c13">I</font> = <font color="Maroon" title="c12">I</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">a9</font> =  <a href="commacat.html#K6" title="COMMACAT:func.6">commaComp</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c10">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c10">M</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">C9</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:10_1_1"/><span class="lab"><font color="Green" title="E12">A9</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> holds  <font color="Maroon" title="c15">C9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">k</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c16">D9</font> = <font color="Maroon" title="c11">D9</font>, <font color="Maroon" title="c17">C9</font> = <font color="Maroon" title="c15">C9</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c9">O</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">FG</font> =  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #);<br/>
<a NAME="E13:10_1_1"/><span class="lab"><font color="Green" title="E13">A10</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">k1</font>,<font color="Olive" title="b2">k2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> : <font color="Olive" title="b1">k1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k2</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a>  </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="commacat.html#D8" target="_self" title="COMMACAT:def.8">Def8</a></span>;<br/>
<a NAME="E14:10_1_1"/><span class="lab"><font color="Green" title="E14">A11</font></span>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #)<br/> for <font color="Olive" title="b3">k1</font>, <font color="Olive" title="b4">k2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">k1</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">k2</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Olive" title="b2">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">k1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b4">k2</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b4">k2</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">k1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b4">k2</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">k1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">f</font>, <font color="Maroon" title="c20">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</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">k1</font>, <font color="Olive" title="b2">k2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font>  st <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k1</font> &amp; <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k2</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c19">f</font> holds <br/><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">k1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b2">k2</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b2">k2</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">k1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b2">k2</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">k1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c21">k1</font>, <font color="Maroon" title="c22">k2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</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="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">k1</font> &amp; <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">k2</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c19">f</font> implies <font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">k1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c22">k2</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c22">k2</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">k1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c22">k2</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">k1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:10_1_1_2"/><span class="lab"><font color="Green" title="E15">A12</font></span>: 
( <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">k1</font> &amp; <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">k2</font> )
 <span class="kw">and </span><br/><a NAME="E2:10_1_1_2"/><span class="lab"><font color="Green" title="E16">A13</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c19">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="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">k1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c22">k2</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c22">k2</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">k1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c22">k2</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">k1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/>

<a NAME="E3:10_1_1_2"/><span class="lab"><font color="Green" title="E17">A14</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">k2</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">k1</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E11:10_1_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E1:10_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a></span>;<br/>
<a NAME="E4:10_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A15</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c22">k2</font>,<font color="Maroon" title="c21">k1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:10_1_1"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="txt" href="#E2:10_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a></span>;<br/>
<a NAME="E5:10_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A16</font></span>: 
<font color="Maroon" title="c14">a9</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c22">k2</font>,<font color="Maroon" title="c21">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">k2</font> <a href="commacat.html#K5" title="COMMACAT:func.5">*</a> <font color="Maroon" title="c21">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="commacat.html#D8" target="_self" title="COMMACAT:def.8">Def8</a></span>;<br/>
<a NAME="E6:10_1_1_2"/>
<font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a9</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">g</font>,<font color="Maroon" title="c19">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E4:10_1_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="ref" href="cat_1.html#D4" title="CAT_1:def.4">CAT_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:10_1_1_2"/>
<font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">k1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c22">k2</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c22">k2</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">k1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c22">k2</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">k1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E1:10_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E2:10_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E3:10_1_1_2"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="txt" href="#E5:10_1_1_2"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="ref" href="commacat.html#D7" target="_self" title="COMMACAT:def.7">Def7</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="E15:10_1_1"/><span class="lab"><font color="Green" title="E15">A17</font></span>: 
for <font color="Olive" title="b1">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #) holds <br/> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; ( for <font color="Olive" title="b2">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #)  st  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) &amp; ( for <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #)  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b2">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> &amp; ( for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #)  st  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> ) &amp; ( for <font color="Olive" title="b1">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #)  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> holds <br/><font color="Olive" title="b1">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c20">b9</font> = <font color="Maroon" title="c19">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">i</font> =  <a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> ;<br/>
<a NAME="E3:10_1_1_3"/><span class="lab"><font color="Green" title="E16">A18</font></span>: 
<font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">b9</font>,<font color="Maroon" title="c20">b9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">b9</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">b9</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10_1_1"><span class="lab"><font color="Green" title="E11">A8</font></span></a></span>;<br/>
<a NAME="E4:10_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A19</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">b9</font></span>)</span> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/>
<a NAME="E5:10_1_1_3"/><span class="lab"><font color="Green" title="E18">A20</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">b9</font></span>)</span> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_1_3"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:10_1_1_3"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E11:10_1_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E4:10_1_1_3"><span class="lab"><font color="Green" title="E17">A19</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"> ( ( for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #)  st  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> ) &amp; ( for <font color="Olive" title="b1">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #)  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> holds <br/><font color="Olive" title="b1">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> ) )</span><br/>

<a NAME="E7:10_1_1_3"/><span class="lab"><font color="Green" title="E19">A21</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">b9</font></span>)</span> <a href="mcart_1.html#K19" title="MCART_1:func.19">`21</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">b9</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">b9</font></span>)</span> <a href="mcart_1.html#K20" title="MCART_1:func.20">`22</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">b9</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_1_3"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/>
<span class="kw">thus </span><a NAME="E8:10_1_1_3"/>
for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #)  st  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</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">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #)  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> holds <br/><font color="Olive" title="b1">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c22">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> implies <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c22">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">f</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c23">f9</font> = <font color="Maroon" title="c22">f</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> ;<br/>
<span class="kw">assume </span><a NAME="E2:10_1_1_3_1"/><span class="lab"><font color="Green" title="E20">A22</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c22">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font>
 ; <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="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c22">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">f</font></span><br/>

<a NAME="E3:10_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A23</font></span>: 
<font color="Maroon" title="c19">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">f</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10_1_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a></span>;<br/>
<a NAME="E4:10_1_1_3_1"/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">f9</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">f9</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a> </span>)</span> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="commacat.html#T3" target="_self" title="COMMACAT:th.3">Th3</a></span>;<br/>
<a NAME="E5:10_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A24</font></span>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">b9</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">f9</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f9</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_1_3_1"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="ref" href="cat_1.html#T46" title="CAT_1:th.46">CAT_1:46</a></span>;<br/>
<a NAME="E6:10_1_1_3_1"/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">f9</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">f9</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a> </span>)</span> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="commacat.html#T3" target="_self" title="COMMACAT:th.3">Th3</a></span>;<br/>
<a NAME="E7:10_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A25</font></span>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">b9</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">f9</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f9</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_1_3_1"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="ref" href="cat_1.html#T46" title="CAT_1:th.46">CAT_1:46</a></span>;<br/>
<a NAME="E8:10_1_1_3_1"/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c22">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E4:10_1_1_3"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E2:10_1_1_3_1"><span class="lab"><font color="Green" title="E20">A22</font></span></a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c22">f</font> = 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">f9</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">f9</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">f9</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:10_1_1"><span class="lab"><font color="Green" title="E14">A11</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c22">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E5:10_1_1_3"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E7:10_1_1_3"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:10_1_1_3_1"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="txt" href="#E7:10_1_1_3_1"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="txt" href="#E5:10_1_1_3_1"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="commacat.html#T3" target="_self" title="COMMACAT:th.3">Th3</a></span>
;<br/> <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>

<span class="kw">let </span><font color="Maroon" title="c22">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> implies <font color="Maroon" title="c22">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c23">g9</font> = <font color="Maroon" title="c22">g</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> ;<br/>
<span class="kw">assume </span><a NAME="E10:10_1_1_3"/><span class="lab"><font color="Green" title="E20">A26</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</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="c22">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font></span><br/>

<a NAME="E11:10_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A27</font></span>: 
<font color="Maroon" title="c19">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>;<br/>
<a NAME="E12:10_1_1_3"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g9</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g9</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a> </span>)</span> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="commacat.html#T3" target="_self" title="COMMACAT:th.3">Th3</a></span>;<br/>
<a NAME="E13:10_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A28</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">g9</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">b9</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">g9</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10_1_1_3"><span class="lab"><font color="Green" title="E21">A27</font></span></a>, <a class="ref" href="cat_1.html#T47" title="CAT_1:th.47">CAT_1:47</a></span>;<br/>
<a NAME="E14:10_1_1_3"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g9</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g9</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a> </span>)</span> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="commacat.html#T3" target="_self" title="COMMACAT:th.3">Th3</a></span>;<br/>
<a NAME="E15:10_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A29</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">g9</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">b9</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">g9</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10_1_1_3"><span class="lab"><font color="Green" title="E21">A27</font></span></a>, <a class="ref" href="cat_1.html#T47" title="CAT_1:th.47">CAT_1:47</a></span>;<br/>
<a NAME="E16:10_1_1_3"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10_1_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E5:10_1_1_3"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:10_1_1_3"><span class="lab"><font color="Green" title="E20">A26</font></span></a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c22">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c19">b</font></span>)</span> = 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c23">g9</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c23">g9</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c23">g9</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:10_1_1"><span class="lab"><font color="Green" title="E14">A11</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c22">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E4:10_1_1_3"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E7:10_1_1_3"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E11:10_1_1_3"><span class="lab"><font color="Green" title="E21">A27</font></span></a>, <a class="txt" href="#E15:10_1_1_3"><span class="lab"><font color="Green" title="E23">A29</font></span></a>, <a class="txt" href="#E13:10_1_1_3"><span class="lab"><font color="Green" title="E22">A28</font></span></a>, <a class="ref" href="commacat.html#T3" target="_self" title="COMMACAT:th.3">Th3</a></span>
;<br/> <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="E16:10_1_1"/><span class="lab"><font color="Green" title="E16">A30</font></span>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #)  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> holds <br/>(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">f</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b2">g</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">f</font>, <font color="Maroon" title="c20">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c19">f</font> implies (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c19">f</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">g</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:10_1_1_4"/><span class="lab"><font color="Green" title="E17">A31</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c19">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c19">f</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">g</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c21">f1</font> = <font color="Maroon" title="c19">f</font>, <font color="Maroon" title="c22">g1</font> = <font color="Maroon" title="c20">g</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> ;<br/>
<a NAME="E3:10_1_1_4"/><span class="lab"><font color="Green" title="E18">A32</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f1</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E11:10_1_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a></span>;<br/>
<a NAME="E4:10_1_1_4"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c22">g1</font>,<font color="Maroon" title="c21">f1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:10_1_1"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="txt" href="#E1:10_1_1_4"><span class="lab"><font color="Green" title="E17">A31</font></span></a></span>;<br/>
<a NAME="E5:10_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A33</font></span>: 
( <font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a9</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">g</font>,<font color="Maroon" title="c19">f</font> &amp; <font color="Maroon" title="c14">a9</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c22">g1</font>,<font color="Maroon" title="c21">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g1</font> <a href="commacat.html#K5" title="COMMACAT:func.5">*</a> <font color="Maroon" title="c21">f1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="commacat.html#D8" target="_self" title="COMMACAT:def.8">Def8</a>, <a class="ref" href="cat_1.html#D4" title="CAT_1:def.4">CAT_1:def 4</a></span>;<br/>
<a NAME="E6:10_1_1_4"/><span class="lab"><font color="Green" title="E20">A34</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f</font> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E11:10_1_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a></span>;<br/>
<a NAME="E7:10_1_1_4"/><span class="lab"><font color="Green" title="E21">A35</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E11:10_1_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a></span>;<br/>
<a NAME="E8:10_1_1_4"/>
<font color="Maroon" title="c22">g1</font> <a href="commacat.html#K5" title="COMMACAT:func.5">*</a> <font color="Maroon" title="c21">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">f1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c22">g1</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c22">g1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">f1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c22">g1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">f1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E1:10_1_1_4"><span class="lab"><font color="Green" title="E17">A31</font></span></a>, <a class="txt" href="#E3:10_1_1_4"><span class="lab"><font color="Green" title="E18">A32</font></span></a>, <a class="ref" href="commacat.html#D7" target="_self" title="COMMACAT:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:10_1_1_4"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c19">f</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10_1_1_4"><span class="lab"><font color="Green" title="E21">A35</font></span></a>, <a class="txt" href="#E6:10_1_1_4"><span class="lab"><font color="Green" title="E20">A34</font></span></a>, <a class="txt" href="#E5:10_1_1_4"><span class="lab"><font color="Green" title="E19">A33</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</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="E17:10_1_1"/><span class="lab"><font color="Green" title="E17">A36</font></span>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #)  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b2">g</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Olive" title="b3">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">f</font>, <font color="Maroon" title="c20">g</font>, <font color="Maroon" title="c21">h</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c21">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">g</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c19">f</font> implies <font color="Maroon" title="c21">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c20">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c22">f1</font> = <font color="Maroon" title="c19">f</font>, <font color="Maroon" title="c23">g1</font> = <font color="Maroon" title="c20">g</font>, <font color="Maroon" title="c24">h1</font> = <font color="Maroon" title="c21">h</font>, <font color="Maroon" title="c25">gf</font> = <font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font>, <font color="Maroon" title="c26">hg</font> = <font color="Maroon" title="c21">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c20">g</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> ;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:10_1_1_5"/><span class="lab"><font color="Green" title="E18">A37</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c21">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">g</font>
 <span class="kw">and </span><br/><a NAME="E3:10_1_1_5"/><span class="lab"><font color="Green" title="E19">A38</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c19">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="c21">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c20">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font></span><br/>

<a NAME="E4:10_1_1_5"/><span class="lab"><font color="Green" title="E20">A39</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g</font> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E11:10_1_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a></span>;<br/>
<a NAME="E5:10_1_1_5"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c20">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:10_1_1"><span class="lab"><font color="Green" title="E16">A30</font></span></a>, <a class="txt" href="#E2:10_1_1_5"><span class="lab"><font color="Green" title="E18">A37</font></span></a></span>;<br/>
<a NAME="E6:10_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A40</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c20">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">f</font> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c26">hg</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c26">hg</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">f1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c26">hg</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">f1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:10_1_1"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="txt" href="#E3:10_1_1_5"><span class="lab"><font color="Green" title="E19">A38</font></span></a></span>;<br/>
<a NAME="E7:10_1_1_5"/><span class="lab"><font color="Green" title="E22">A41</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c21">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h</font> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E11:10_1_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a></span>;<br/>
<a NAME="E8:10_1_1_5"/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c20">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:10_1_1"><span class="lab"><font color="Green" title="E16">A30</font></span></a>, <a class="txt" href="#E3:10_1_1_5"><span class="lab"><font color="Green" title="E19">A38</font></span></a></span>;<br/>
<a NAME="E9:10_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A42</font></span>: 
<font color="Maroon" title="c21">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c25">gf</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c21">h</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">gf</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">gf</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:10_1_1"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="txt" href="#E2:10_1_1_5"><span class="lab"><font color="Green" title="E18">A37</font></span></a></span>;<br/>
<a NAME="E10:10_1_1_5"/><span class="lab"><font color="Green" title="E24">A43</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a> </span>)</span> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">f1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">f1</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a> </span>)</span> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="commacat.html#T3" target="_self" title="COMMACAT:th.3">Th3</a></span>;<br/>
<a NAME="E11:10_1_1_5"/><span class="lab"><font color="Green" title="E25">A44</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a> </span>)</span> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">f1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">f1</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a> </span>)</span> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="commacat.html#T3" target="_self" title="COMMACAT:th.3">Th3</a></span>;<br/>
<a NAME="E12:10_1_1_5"/><span class="lab"><font color="Green" title="E26">A45</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a> </span>)</span> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a> </span>)</span> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="commacat.html#T3" target="_self" title="COMMACAT:th.3">Th3</a></span>;<br/>
<a NAME="E13:10_1_1_5"/><span class="lab"><font color="Green" title="E27">A46</font></span>: 
<font color="Maroon" title="c21">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c21">h</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:10_1_1"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="txt" href="#E2:10_1_1_5"><span class="lab"><font color="Green" title="E18">A37</font></span></a></span>;<br/>
<a NAME="E14:10_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A47</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c21">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c20">g</font></span>)</span> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  &amp; <font color="Maroon" title="c26">hg</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/>
<a NAME="E15:10_1_1_5"/><span class="lab"><font color="Green" title="E29">A48</font></span>: 
<font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">f</font> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">f1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">f1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:10_1_1"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="txt" href="#E3:10_1_1_5"><span class="lab"><font color="Green" title="E19">A38</font></span></a></span>;<br/>
<a NAME="E16:10_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A49</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f</font> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a>  &amp; <font color="Maroon" title="c25">gf</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">f1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/>
<a NAME="E17:10_1_1_5"/><span class="lab"><font color="Green" title="E31">A50</font></span>: 
<font color="Maroon" title="c25">gf</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">f1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:10_1_1_5"><span class="lab"><font color="Green" title="E29">A48</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/>
<a NAME="E18:10_1_1_5"/><span class="lab"><font color="Green" title="E32">A51</font></span>: 
<font color="Maroon" title="c26">hg</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:10_1_1_5"><span class="lab"><font color="Green" title="E27">A46</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/>
<a NAME="E19:10_1_1_5"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a> </span>)</span> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a> </span>)</span> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="commacat.html#T3" target="_self" title="COMMACAT:th.3">Th3</a></span>;<br/>
<a NAME="E20:10_1_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">f1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">f1</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_1_1_5"><span class="lab"><font color="Green" title="E18">A37</font></span></a>, <a class="txt" href="#E3:10_1_1_5"><span class="lab"><font color="Green" title="E19">A38</font></span></a>, <a class="txt" href="#E4:10_1_1_5"><span class="lab"><font color="Green" title="E20">A39</font></span></a>, <a class="txt" href="#E10:10_1_1_5"><span class="lab"><font color="Green" title="E24">A43</font></span></a>, <a class="txt" href="#E7:10_1_1_5"><span class="lab"><font color="Green" title="E22">A41</font></span></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:10_1_1_5"/>
<font color="Maroon" title="c21">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">h</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c20">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c19">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_1_1_5"><span class="lab"><font color="Green" title="E18">A37</font></span></a>, <a class="txt" href="#E3:10_1_1_5"><span class="lab"><font color="Green" title="E19">A38</font></span></a>, <a class="txt" href="#E4:10_1_1_5"><span class="lab"><font color="Green" title="E20">A39</font></span></a>, <a class="txt" href="#E7:10_1_1_5"><span class="lab"><font color="Green" title="E22">A41</font></span></a>, <a class="txt" href="#E12:10_1_1_5"><span class="lab"><font color="Green" title="E26">A45</font></span></a>, <a class="txt" href="#E11:10_1_1_5"><span class="lab"><font color="Green" title="E25">A44</font></span></a>, <a class="txt" href="#E9:10_1_1_5"><span class="lab"><font color="Green" title="E23">A42</font></span></a>, <a class="txt" href="#E6:10_1_1_5"><span class="lab"><font color="Green" title="E21">A40</font></span></a>, <a class="txt" href="#E14:10_1_1_5"><span class="lab"><font color="Green" title="E28">A47</font></span></a>, <a class="txt" href="#E16:10_1_1_5"><span class="lab"><font color="Green" title="E30">A49</font></span></a>, <a class="txt" href="#E18:10_1_1_5"><span class="lab"><font color="Green" title="E32">A51</font></span></a>, <a class="txt" href="#E17:10_1_1_5"><span class="lab"><font color="Green" title="E31">A50</font></span></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</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="E18:10_1_1"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #) holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #) iff  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">f</font>, <font color="Maroon" title="c20">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">g</font>,<font color="Maroon" title="c19">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #) iff  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c19">f</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c21">f1</font> = <font color="Maroon" title="c19">f</font>, <font color="Maroon" title="c22">g1</font> = <font color="Maroon" title="c20">g</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> ;<br/>
<a NAME="E2:10_1_1_6"/><span class="lab"><font color="Green" title="E18">A52</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">f1</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E11:10_1_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:10_1_1_6"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">g</font>,<font color="Maroon" title="c19">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #) implies  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c19">f</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c19">f</font> implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">g</font>,<font color="Maroon" title="c19">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:10_1_1_6_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">g</font>,<font color="Maroon" title="c19">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c19">f</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">k1</font>, <font color="Maroon" title="c24">k2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font><span class="kw"> such that </span><br/><a NAME="E3:10_1_1_6_1"/><span class="lab"><font color="Green" title="E19">A53</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">g</font>,<font color="Maroon" title="c19">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c23">k1</font>,<font color="Maroon" title="c24">k2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:10_1_1_6_1"/><span class="lab"><font color="Green" title="E20">A54</font></span>: 
<font color="Maroon" title="c23">k1</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">k2</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:10_1_1"><span class="lab"><font color="Green" title="E13">A10</font></span></a></span>;<br/>
<a NAME="E5:10_1_1_6_1"/>
<font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_1_6_1"><span class="lab"><font color="Green" title="E19">A53</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:10_1_1_6_1"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c19">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_1_1_6"><span class="lab"><font color="Green" title="E18">A52</font></span></a>, <a class="txt" href="#E3:10_1_1_6_1"><span class="lab"><font color="Green" title="E19">A53</font></span></a>, <a class="txt" href="#E4:10_1_1_6_1"><span class="lab"><font color="Green" title="E20">A54</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</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>

<span class="kw">thus </span><a NAME="E4:10_1_1_6"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c19">f</font> implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">g</font>,<font color="Maroon" title="c19">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:10_1_1"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="txt" href="#E2:10_1_1_6"><span class="lab"><font color="Green" title="E18">A52</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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">FG</font> =  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c9">O</font>,<font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c16">D9</font>,<font color="Maroon" title="c17">C9</font>,<font color="Maroon" title="c14">a9</font>,<font color="Maroon" title="c13">I</font> #) as   <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a>  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:10_1_1"><span class="lab"><font color="Green" title="E16">A30</font></span></a>, <a class="txt" href="#E17:10_1_1"><span class="lab"><font color="Green" title="E17">A36</font></span></a>, <a class="txt" href="#E15:10_1_1"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="ref" href="cat_1.html#T29" title="CAT_1:th.29">CAT_1:29</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c19">FG</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c19">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> &amp; the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c19">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> holds  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c19">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a>  ) &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> holds  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c19">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a>  ) &amp; ( for <font color="Olive" title="b1">o</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> holds  the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c19">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Olive" title="b1">o</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c19">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="commacat.html#K6" title="COMMACAT:func.6">commaComp</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> )</span><br/>

<span class="kw">thus </span><a NAME="E20:10_1_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c19">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> &amp; the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c19">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> holds  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c19">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="commacat.html#K3" title="COMMACAT:func.3">`11</a>  ) &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> holds  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c19">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="commacat.html#K4" title="COMMACAT:func.4">`12</a>  ) &amp; ( for <font color="Olive" title="b1">o</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> holds  the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c19">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Olive" title="b1">o</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c19">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="commacat.html#K6" title="COMMACAT:func.6">commaComp</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E11:10_1_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E7:10_1_1"><span class="lab"><font color="Green" title="E11">A8</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/>


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