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<div class="add">

<b>deffunc </b><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>) <b>-&gt; </b>   <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/>
<b>deffunc </b><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>) <b>-&gt; </b>   <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/>
<b>consider </b><font color="Maroon" title="c9">D'</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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 href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font><b> such that </b><br/><a NAME="E2:9_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
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="c9">D'</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>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<b>consider </b><font color="Maroon" title="c10">C'</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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 href="commacat.html#K1" title="COMMACAT:func.1">commaObjs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font><b> such that </b><br/><a NAME="E4:9_1_1"/><i><font color="Green" title="E9">A3</font></i>: 
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="c10">C'</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>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E5:9_1_1"/><i><font color="Green" title="E10">A4</font></i>: 
(  <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#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="b1">c</font>,<font color="Olive" title="b2">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b3">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">c</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>, <font color="Olive" title="b2">d</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font>, <font color="Olive" title="b3">f</font> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">E</font> : <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Olive" title="b1">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Olive" title="b2">d</font></span>)</span> </span>}</span>  &amp;  <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="cat_1.html#K3" title="CAT_1:func.3">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="cat_1.html#K4" title="CAT_1:func.4">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="cat_1.html#K3" title="CAT_1:func.3">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="cat_1.html#K4" title="CAT_1:func.4">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#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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>  )
 
<b>by </b><i><a class="txt" href="commacat.html#E2:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="commacat.html#D5" target="_self" title="COMMACAT:def.5">Def5</a>, <a class="ref" href="commacat.html#D6" target="_self" title="COMMACAT:def.6">Def6</a></i>;<br/>
<b>defpred </b><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> ] <b>means </b>$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#K10" title="CAT_1:func.10">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#K10" title="CAT_1:func.10">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/>
<a NAME="E6:9_1_1"/><i><font color="Green" title="E11">A5</font></i>: 
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="9_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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>;<br/>

<a NAME="E1:9_1_1_1"/><i><font color="Green" title="E12">A6</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">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="c11">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a>  &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">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="c11">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">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="c11">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a>  &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">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="c11">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a>  &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#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">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#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">o</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></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#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">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#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">o</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="commacat.html#E2:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="commacat.html#T2" target="_self" title="COMMACAT:th.2">Th2</a>, <a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a>, <a class="ref" href="cat_1.html#T108" title="CAT_1:th.108">CAT_1:108</a></i>;<br/>
<a NAME="E2:9_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c11">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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#K10" title="CAT_1:func.10">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">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="c11">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  &amp; <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#K10" title="CAT_1:func.10">id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">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#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">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="c11">o</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  )
 
<b>by </b><i><a class="ref" href="cat_1.html#T46" title="CAT_1:th.46">CAT_1:46</a>, <a class="ref" href="cat_1.html#T47" title="CAT_1:th.47">CAT_1:47</a></i>;<br/>
<a NAME="E3:9_1_1_1"/><b>then </b>
<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="c11">o</font>,<font color="Maroon" title="c11">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#K10" title="CAT_1:func.10">id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">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#K10" title="CAT_1:func.10">id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">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>
 
<b>by </b><i><a class="txt" href="commacat.html#E5:9_1_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="commacat.html#E1:9_1_1_1"><i><font color="Green" title="E12">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:9_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="c11">o</font>,<font color="Olive" title="b1">k</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c11">I</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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="commacat.html#K2" title="COMMACAT:func.2">commaMorphs</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font><b> such that </b><br/><a NAME="E8:9_1_1"/><i><font color="Green" title="E12">A7</font></i>: 
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="c11">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">o</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="commacat.html#E6:9_1_1"><i><font color="Green" title="E11">A5</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c12">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="c13">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/>
<b>reconsider </b><font color="Maroon" title="c14">D'</font> = <font color="Maroon" title="c9">D'</font>, <font color="Maroon" title="c15">C'</font> = <font color="Maroon" title="c10">C'</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c12">O</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">'</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="c13">M</font>,<font color="Maroon" title="c13">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c13">M</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">I</font> = <font color="Maroon" title="c11">I</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c12">O</font>,<font color="Maroon" title="c13">M</font> ;<br/>
<b>set </b><font color="Maroon" title="c18">FG</font> =  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">I</font> #);<br/>
<a NAME="E13:9_1_1"/><i><font color="Green" title="E13">A8</font></i>: 
 <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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">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> 
 
<b>by </b><i><a class="ref" href="commacat.html#D8" target="_self" title="COMMACAT:def.8">Def8</a></i>;<br/>
<a NAME="E14:9_1_1"/><i><font color="Green" title="E14">A9</font></i>: 
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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">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="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Olive" title="b2">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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="9_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">I</font> #);<br/><b>let </b><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>;<br/>





<b>assume </b><a NAME="E1:9_1_1_2"/><i><font color="Green" title="E15">A10</font></i>: 
( <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="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c19">f</font> )
 ;<br/>

<a NAME="E2:9_1_1_2"/><b>then </b><i><font color="Green" title="E16">A11</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">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="cat_1.html#K4" title="CAT_1:func.4">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>  )
 
<b>by </b><i><a class="txt" href="commacat.html#E2:9_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="commacat.html#E4:9_1_1"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/>
<a NAME="E3:9_1_1_2"/><b>then </b>
<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="c16">'</font>
 
<b>by </b><i><a class="txt" href="commacat.html#E13:9_1_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="commacat.html#E1:9_1_1_2"><i><font color="Green" title="E15">A10</font></i></a></i>;<br/>
<a NAME="E4:9_1_1_2"/><b>then </b>
( <font color="Maroon" title="c20">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">'</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="c16">'</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> &amp; <font color="Maroon" title="c22">k2</font> <a href="commacat.html#K5" title="COMMACAT:func.5">*</a> <font color="Maroon" title="c21">k1</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#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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> )
 
<b>by </b><i><a class="txt" href="commacat.html#E2:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="commacat.html#E1:9_1_1_2"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="commacat.html#E2:9_1_1_2"><i><font color="Green" title="E16">A11</font></i></a>, <a class="ref" href="commacat.html#D7" target="_self" title="COMMACAT:def.7">Def7</a>, <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></i>;<br/>
<b>hence </b><a NAME="E5:9_1_1_2"/>
<font color="Maroon" title="c20">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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>
 <b>by </b><i><a class="txt" href="commacat.html#E1:9_1_1_2"><i><font color="Green" title="E15">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:9_1_1"/><i><font color="Green" title="E15">A12</font></i>: 
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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">I</font> #) iff  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b1">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">I</font> #);<br/>

<b>reconsider </b><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="E2:9_1_1_3"/><i><font color="Green" title="E16">A13</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">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="cat_1.html#K4" title="CAT_1:func.4">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>  )
 
<b>by </b><i><a class="txt" href="commacat.html#E2:9_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="commacat.html#E4:9_1_1"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/>
<b>thus </b><a NAME="E3:9_1_1_3"/>
( <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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">I</font> #) implies  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c19">f</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1_1_3_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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">I</font> #)
 ;<br/>

<b>then consider </b><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><b> such that </b><br/><a NAME="E3:9_1_1_3_1"/><i><font color="Green" title="E17">A14</font></i>: 
( <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> &amp; <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>  )
 <b>by </b><i><a class="txt" href="commacat.html#E13:9_1_1"><i><font color="Green" title="E13">A8</font></i></a></i>;<br/>
<a NAME="E4:9_1_1_3_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> &amp; <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">k2</font> )
 
<b>by </b><i><a class="txt" href="commacat.html#E3:9_1_1_3_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E5:9_1_1_3_1"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c19">f</font>
 <b>by </b><i><a class="txt" href="commacat.html#E2:9_1_1_3"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="commacat.html#E3:9_1_1_3_1"><i><font color="Green" title="E17">A14</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E4:9_1_1_3"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">I</font> #) )
 <b>by </b><i><a class="txt" href="commacat.html#E13:9_1_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="commacat.html#E2:9_1_1_3"><i><font color="Green" title="E16">A13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:9_1_1"/><i><font color="Green" title="E16">A15</font></i>: 
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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">I</font> #)  st  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b1">f</font> holds <br/>(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b1">f</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b2">g</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">I</font> #);<br/>

<b>assume </b><a NAME="E1:9_1_1_4"/><i><font color="Green" title="E17">A16</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c19">f</font>
 ;<br/>

<b>reconsider </b><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="E3:9_1_1_4"/><i><font color="Green" title="E18">A17</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">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="cat_1.html#K4" title="CAT_1:func.4">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>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">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="cat_1.html#K4" title="CAT_1:func.4">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>  )
 
<b>by </b><i><a class="txt" href="commacat.html#E2:9_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="commacat.html#E4:9_1_1"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/>
<a NAME="E4:9_1_1_4"/><b>then </b>
<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="c16">'</font>
 
<b>by </b><i><a class="txt" href="commacat.html#E13:9_1_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="commacat.html#E1:9_1_1_4"><i><font color="Green" title="E17">A16</font></i></a></i>;<br/>
<a NAME="E5:9_1_1_4"/><b>then </b>
( <font color="Maroon" title="c20">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">'</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="c16">'</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> &amp; <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#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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> )
 
<b>by </b><i><a class="txt" href="commacat.html#E2:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="commacat.html#E1:9_1_1_4"><i><font color="Green" title="E17">A16</font></i></a>, <a class="txt" href="commacat.html#E3:9_1_1_4"><i><font color="Green" title="E18">A17</font></i></a>, <a class="ref" href="commacat.html#D7" target="_self" title="COMMACAT:def.7">Def7</a>, <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></i>;<br/>
<b>hence </b><a NAME="E6:9_1_1_4"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c19">f</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c20">g</font> )
 <b>by </b><i><a class="txt" href="commacat.html#E3:9_1_1_4"><i><font color="Green" title="E18">A17</font></i></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:9_1_1"/><i><font color="Green" title="E17">A18</font></i>: 
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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">I</font> #)  st  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b2">g</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Olive" title="b3">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">I</font> #);<br/>

<b>reconsider </b><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#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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/>
<b>assume </b><a NAME="E2:9_1_1_5"/><i><font color="Green" title="E18">A19</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c21">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c20">g</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c19">f</font> )
 ;<br/>

<a NAME="E3:9_1_1_5"/><b>then </b>
( <font color="Maroon" title="c20">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">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; <font color="Maroon" title="c21">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">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>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">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="cat_1.html#K4" title="CAT_1:func.4">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>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">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="cat_1.html#K3" title="CAT_1:func.3">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="cat_1.html#K4" title="CAT_1:func.4">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>  &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">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>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">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="cat_1.html#K4" title="CAT_1:func.4">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>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">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="cat_1.html#K4" title="CAT_1:func.4">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>  &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c19">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c20">g</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c20">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c20">g</font> )
 
<b>by </b><i><a class="txt" href="commacat.html#E2:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="commacat.html#E2:9_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="commacat.html#E4:9_1_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="commacat.html#E14:9_1_1"><i><font color="Green" title="E14">A9</font></i></a>, <a class="txt" href="commacat.html#E16:9_1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="ref" href="commacat.html#T3" target="_self" title="COMMACAT:th.3">Th3</a></i>;<br/>
<a NAME="E4:9_1_1_5"/><b>then </b>
( <font color="Maroon" title="c21">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c21">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c20">g</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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; <span class="p1">(<span class="default"><font color="Maroon" title="c21">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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> &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#K5" title="CAT_1:func.5">*</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> &amp; <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#K5" title="CAT_1:func.5">*</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> &amp; <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#K5" title="CAT_1:func.5">*</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> &amp; <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#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</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> &amp; <span class="p1">(<span class="default"><span class="p2">(<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#K5" title="CAT_1:func.5">*</a> <span class="p2">(<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> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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="c24">h1</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p2">(<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> )
 
<b>by </b><i><a class="txt" href="commacat.html#E14:9_1_1"><i><font color="Green" title="E14">A9</font></i></a>, <a class="txt" href="commacat.html#E2:9_1_1_5"><i><font color="Green" title="E18">A19</font></i></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/>
<b>hence </b><a NAME="E5:9_1_1_5"/>
<font color="Maroon" title="c21">h</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c20">g</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c19">f</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E18:9_1_1"/>
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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">I</font> #) holds <br/> (  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">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="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">I</font> #)  st  <a href="cat_1.html#K4" title="CAT_1:func.4">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#K6" title="CAT_1:func.6">id</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">I</font> #)  st  <a href="cat_1.html#K3" title="CAT_1:func.3">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#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">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="9_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">I</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c20">b'</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/>
<b>reconsider </b><font color="Maroon" title="c21">i</font> =  <a href="cat_1.html#K6" title="CAT_1:func.6">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="E3:9_1_1_6"/>
<font color="Maroon" title="c17">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">b'</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">b'</font>,<font color="Maroon" title="c20">b'</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#K10" title="CAT_1:func.10">id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">b'</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#K10" title="CAT_1:func.10">id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">b'</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>
 
<b>by </b><i><a class="txt" href="commacat.html#E8:9_1_1"><i><font color="Green" title="E12">A7</font></i></a></i>;<br/>
<a NAME="E4:9_1_1_6"/><b>then </b><i><font color="Green" title="E18">A20</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c17">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">b'</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">b'</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">b'</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">b'</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">b'</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#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">b'</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="c17">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">b'</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#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">b'</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span> &amp; <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">b</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/>
<b>hence </b><a NAME="E5:9_1_1_6"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">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="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">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> )
 <b>by </b><i><a class="txt" href="commacat.html#E2:9_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="commacat.html#E4:9_1_1"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:9_1_1_6"/>
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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">I</font> #)  st  <a href="cat_1.html#K4" title="CAT_1:func.4">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#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c19">b</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_6_1"><b>proof </b></a><div class="add">

<b>let </b><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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">I</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c23">f'</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/>
<b>assume </b><a NAME="E2:9_1_1_6_1"/>
 <a href="cat_1.html#K4" title="CAT_1:func.4">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>
 ;<br/>

<a NAME="E3:9_1_1_6_1"/><b>then </b><i><font color="Green" title="E19">A21</font></i>: 
(  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c22">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c19">b</font></span>)</span> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">b'</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="c20">b'</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">b'</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="c20">b'</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a>  &amp; <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>  &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">f'</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">f'</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>  &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">f'</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">f'</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>  )
 
<b>by </b><i><a class="txt" href="commacat.html#E2:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="commacat.html#E2:9_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="commacat.html#E4:9_1_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="commacat.html#E4:9_1_1_6"><i><font color="Green" title="E18">A20</font></i></a>, <a class="ref" href="commacat.html#T3" target="_self" title="COMMACAT:th.3">Th3</a></i>;<br/>
<a NAME="E4:9_1_1_6_1"/><b>then </b><i><font color="Green" title="E20">A22</font></i>: 
( <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">b'</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a> </span>)</span></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">f'</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">f'</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a>  &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">b'</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a> </span>)</span></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">f'</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">f'</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a>  )
 
<b>by </b><i><a class="ref" href="cat_1.html#T46" title="CAT_1:th.46">CAT_1:46</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c19">b</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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">f'</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#K5" title="CAT_1:func.5">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">f'</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#K5" title="CAT_1:func.5">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">f'</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>
<b>by </b><i><a class="txt" href="commacat.html#E14:9_1_1"><i><font color="Green" title="E14">A9</font></i></a>, <a class="txt" href="commacat.html#E3:9_1_1_6_1"><i><font color="Green" title="E19">A21</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c22">f</font>
<b>by </b><i><a class="txt" href="commacat.html#E2:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="commacat.html#E4:9_1_1_6"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="commacat.html#E3:9_1_1_6_1"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="commacat.html#E4:9_1_1_6_1"><i><font color="Green" title="E20">A22</font></i></a>, <a class="ref" href="commacat.html#T3" target="_self" title="COMMACAT:th.3">Th3</a></i>
;<br/>


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

<b>let </b><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="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">I</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c23">g'</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/>
<b>assume </b><a NAME="E8:9_1_1_6"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">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>
 ;<br/>

<a NAME="E9:9_1_1_6"/><b>then </b><i><font color="Green" title="E19">A23</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c19">b</font></span>)</span> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">b'</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="c20">b'</font> <a href="domain_1.html#K15" title="DOMAIN_1:func.15">`12</a>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">b'</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="c20">b'</font> <a href="domain_1.html#K14" title="DOMAIN_1:func.14">`11</a>  &amp; <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>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g'</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">g'</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="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g'</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">g'</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>  )
 
<b>by </b><i><a class="txt" href="commacat.html#E2:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="commacat.html#E2:9_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="commacat.html#E4:9_1_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="commacat.html#E4:9_1_1_6"><i><font color="Green" title="E18">A20</font></i></a>, <a class="ref" href="commacat.html#T3" target="_self" title="COMMACAT:th.3">Th3</a></i>;<br/>
<a NAME="E10:9_1_1_6"/><b>then </b><i><font color="Green" title="E20">A24</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c23">g'</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">b'</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">g'</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c23">g'</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">b'</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">g'</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a>  )
 
<b>by </b><i><a class="ref" href="cat_1.html#T47" title="CAT_1:th.47">CAT_1:47</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c22">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">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">g'</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">g'</font> <a href="domain_1.html#K16" title="DOMAIN_1:func.16">`21</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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">g'</font> <a href="domain_1.html#K17" title="DOMAIN_1:func.17">`22</a> </span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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>
<b>by </b><i><a class="txt" href="commacat.html#E14:9_1_1"><i><font color="Green" title="E14">A9</font></i></a>, <a class="txt" href="commacat.html#E9:9_1_1_6"><i><font color="Green" title="E19">A23</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c22">g</font>
<b>by </b><i><a class="txt" href="commacat.html#E2:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="commacat.html#E4:9_1_1_6"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="commacat.html#E9:9_1_1_6"><i><font color="Green" title="E19">A23</font></i></a>, <a class="txt" href="commacat.html#E10:9_1_1_6"><i><font color="Green" title="E20">A24</font></i></a>, <a class="ref" href="commacat.html#T3" target="_self" title="COMMACAT:th.3">Th3</a></i>
;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c19">FG</font> =  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c12">O</font>,<font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c14">D'</font>,<font color="Maroon" title="c15">C'</font>,<font color="Maroon" title="c16">'</font>,<font color="Maroon" title="c17">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> <b>by </b><i><a class="txt" href="commacat.html#E15:9_1_1"><i><font color="Green" title="E15">A12</font></i></a>, <a class="txt" href="commacat.html#E16:9_1_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="commacat.html#E17:9_1_1"><i><font color="Green" title="E17">A18</font></i></a>, <a class="ref" href="cat_1.html#T29" title="CAT_1:th.29">CAT_1:29</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c19">FG</font>
;<br/>

<b>thus </b><a NAME="E20:9_1_1"/>
( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</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="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</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#U3" title="GRAPH_1:sel.3">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#U4" title="GRAPH_1:sel.4">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#K10" title="CAT_1:func.10">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#K10" title="CAT_1:func.10">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> )
 <b>by </b><i><a class="txt" href="commacat.html#E2:9_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="commacat.html#E4:9_1_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="commacat.html#E8:9_1_1"><i><font color="Green" title="E12">A7</font></i></a></i>;<br/>


</div>
