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

<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> = <font color="Maroon">a<sub>1</sub></font> <a href="commacat.html#K3" target="_self">`11</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> = <font color="Maroon">a<sub>1</sub></font> <a href="commacat.html#K4" target="_self">`12</a> ;<br/>
<b>consider </b><font color="Maroon">D'</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>, <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:9_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> holds  <font color="Maroon">D'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>consider </b><font color="Maroon">C'</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>, <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:9_1_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> holds  <font color="Maroon">C'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E5:9_1_1"/><i><font color="Green">E50</font></i>: 
(  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">c</font>,<font color="Olive">d</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">c</font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Olive">d</font> is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font>, <font color="Olive">f</font> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">E</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Olive">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <font color="Olive">d</font></span>)</span> </span>}</span>  &amp;  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">o1</font>,<font color="Olive">o2</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">h</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">g</font> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>, <font color="Olive">h</font> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">D</font>, <font color="Olive">o1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>, <font color="Olive">o2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> : (  <a href="cat_1.html#K2">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">o1</font> <a href="domain_1.html#K14">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">o2</font> <a href="domain_1.html#K14">`11</a>  &amp;  <a href="cat_1.html#K2">dom</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">o1</font> <a href="domain_1.html#K15">`12</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">o2</font> <a href="domain_1.html#K15">`12</a>  &amp; <span class="p1">(<span class="default"><font color="Olive">o2</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Olive">o1</font> <a href="domain_1.html#K3">`2</a> </span>)</span> ) </span>}</span>  )
 
<b>by </b><i>, , </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="domain_1.html#K14">`11</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="domain_1.html#K15">`12</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span>;<br/>
<a NAME="E6:9_1_1"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">o</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">o</font>,<font color="Olive">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">o</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>;<br/>

<a NAME="E1:9_1_1_1"/><i><font color="Green">E52</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="domain_1.html#K14">`11</a> </span>)</span> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="domain_1.html#K15">`12</a> </span>)</span> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="domain_1.html#K14">`11</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="domain_1.html#K14">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="domain_1.html#K14">`11</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="domain_1.html#K14">`11</a>  &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="domain_1.html#K15">`12</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="domain_1.html#K15">`12</a>  &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="domain_1.html#K15">`12</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="domain_1.html#K15">`12</a>  &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="domain_1.html#K14">`11</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="domain_1.html#K14">`11</a> </span>)</span></span>)</span> &amp; <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="domain_1.html#K15">`12</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">o</font> <a href="domain_1.html#K15">`12</a> </span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="cat_1.html#T44">CAT_1:44</a>, <a class="ref" href="cat_1.html#T108">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">o</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><font color="Maroon">o</font> <a href="domain_1.html#K14">`11</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="domain_1.html#K3">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><font color="Maroon">o</font> <a href="domain_1.html#K15">`12</a> </span>)</span></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="domain_1.html#K3">`2</a>  )
 
<b>by </b><i><a class="ref" href="cat_1.html#T46">CAT_1:46</a>, <a class="ref" href="cat_1.html#T47">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">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">o</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="domain_1.html#K14">`11</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><font color="Maroon">o</font> <a href="domain_1.html#K15">`12</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:9_1_1_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">o</font>,<font color="Olive">k</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">I</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>, <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font><b> such that </b><br/><a NAME="E8:9_1_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">o</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">o</font>,<font color="Maroon">I</font> <a href="funct_2.html#K3">.</a> <font color="Olive">o</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">O</font> =  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>, <font color="Maroon">M</font> =  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">D'</font> = <font color="Maroon">D'</font>, <font color="Maroon">C'</font> = <font color="Maroon">C'</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">M</font>,<font color="Maroon">O</font> ;<br/>
<b>reconsider </b><font color="Maroon">'</font> =  <a href="commacat.html#K6" target="_self">commaComp</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> as   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">M</font> ;<br/>
<b>reconsider </b><font color="Maroon">I</font> = <font color="Maroon">I</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">O</font>,<font color="Maroon">M</font> ;<br/>
<b>set </b><font color="Maroon">FG</font> =  <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #);<br/>
<a NAME="E13:9_1_1"/><i><font color="Green">E54</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #) <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">k1</font>,<font color="Olive">k2</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">k1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>, <font color="Olive">k2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> : <font color="Olive">k1</font> <a href="commacat.html#K3" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Olive">k2</font> <a href="commacat.html#K4" target="_self">`12</a>  </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E14:9_1_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">f</font> being  <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #) holds <br/> (  <a href="cat_1.html#K2">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="mcart_1.html#K17">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="mcart_1.html#K18">`12</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:9_1_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #)<br/> for <font color="Olive">k1</font>, <font color="Olive">k2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">k1</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">k2</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> holds <br/><font color="Olive">g</font> <a href="cat_1.html#K4">*</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive">k1</font> <a href="commacat.html#K3" target="_self">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Olive">k2</font> <a href="commacat.html#K4" target="_self">`12</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">k2</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Olive">k1</font> <a href="domain_1.html#K16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">k2</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Olive">k1</font> <a href="domain_1.html#K17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</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">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #), <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #);<br/><b>let </b><font color="Maroon">k1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">k2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>;<br/>





<b>assume </b><a NAME="E1:9_1_1_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E2:9_1_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font> <a href="commacat.html#K3" target="_self">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> <a href="commacat.html#K4" target="_self">`12</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E3:9_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">k2</font>,<font color="Maroon">k1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">'</font>
 
<b>by </b><i>, <a class="txt" href="commacat.html#E6"><i><font color="Green">Lemma17</font></i></a></i>;<br/>
<a NAME="E4:9_1_1_2"/><b>then </b>
( <font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">'</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">'</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">k2</font>,<font color="Maroon">k1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font> <a href="commacat.html#K5" target="_self">*</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">k2</font> <a href="commacat.html#K5" target="_self">*</a> <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k1</font> <a href="commacat.html#K3" target="_self">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">k2</font> <a href="commacat.html#K4" target="_self">`12</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k2</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="domain_1.html#K16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k2</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="domain_1.html#K17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i>, <a class="txt" href="commacat.html#E6"><i><font color="Green">Lemma17</font></i></a>, , , , <a class="ref" href="cat_1.html#D4">CAT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:9_1_1_2"/>
<font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k1</font> <a href="commacat.html#K3" target="_self">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">k2</font> <a href="commacat.html#K4" target="_self">`12</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k2</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="domain_1.html#K16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k2</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">k1</font> <a href="domain_1.html#K17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="commacat.html#E6"><i><font color="Green">Lemma17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:9_1_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #) holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #) iff  <a href="cat_1.html#K2">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">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">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #), <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #);<br/>

<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font>, <font color="Maroon">g1</font> = <font color="Maroon">g</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> ;<br/>
<a NAME="E2:9_1_1_3"/><i><font color="Green">E71</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="commacat.html#K3" target="_self">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="commacat.html#K4" target="_self">`12</a>  )
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:9_1_1_3"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #) implies  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">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">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #)
 ;<br/>

<b>then consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>, <font color="Maroon">k2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:9_1_1_3_1"/><i><font color="Green">E72</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">k1</font>,<font color="Maroon">k2</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">k1</font> <a href="commacat.html#K3" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font> <a href="commacat.html#K4" target="_self">`12</a>  )
 <b>by </b><i/>;<br/>
<a NAME="E4:9_1_1_3_1"/>
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E5:9_1_1_3_1"/>
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E4:9_1_1_3"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> implies <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:9_1_1"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #)  st  <a href="cat_1.html#K2">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> holds <br/>(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="cat_1.html#K4">*</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Olive">f</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="cat_1.html#K4">*</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">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">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #), <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #);<br/>

<b>assume </b><a NAME="E1:9_1_1_4"/><i><font color="Green">E74</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font>, <font color="Maroon">g1</font> = <font color="Maroon">g</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> ;<br/>
<a NAME="E3:9_1_1_4"/><i><font color="Green">E75</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="commacat.html#K3" target="_self">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="commacat.html#K4" target="_self">`12</a>  &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K17">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K18">`12</a>  &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mcart_1.html#K17">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="mcart_1.html#K18">`12</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E4:9_1_1_4"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">f1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">'</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:9_1_1_4"/><b>then </b>
( <font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">'</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">'</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">f1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="commacat.html#K5" target="_self">*</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">g1</font> <a href="commacat.html#K5" target="_self">*</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="commacat.html#K3" target="_self">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g1</font> <a href="commacat.html#K4" target="_self">`12</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g1</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="domain_1.html#K16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g1</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="domain_1.html#K17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i>, , , , , <a class="ref" href="cat_1.html#D4">CAT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:9_1_1_4"/>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:9_1_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font>, <font color="Olive">h</font> being  <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #)  st  <a href="cat_1.html#K2">dom</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">g</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> holds <br/><font color="Olive">h</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="cat_1.html#K4">*</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="cat_1.html#K4">*</a> <font color="Olive">g</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Olive">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">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #), <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #), <font color="Maroon">h</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #);<br/>

<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font>, <font color="Maroon">g1</font> = <font color="Maroon">g</font>, <font color="Maroon">h1</font> = <font color="Maroon">h</font>, <font color="Maroon">gf</font> = <font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font>, <font color="Maroon">hg</font> = <font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> ;<br/>
<b>assume </b><a NAME="E2:9_1_1_5"/><i><font color="Green">E80</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E3:9_1_1_5"/><b>then </b>
( <font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K17">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K18">`12</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g1</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="domain_1.html#K16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g1</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="domain_1.html#K17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="mcart_1.html#K17">`11</a>  &amp; <font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K17">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="mcart_1.html#K18">`12</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h1</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">g1</font> <a href="domain_1.html#K16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h1</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">g1</font> <a href="domain_1.html#K17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="mcart_1.html#K18">`12</a>  &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="commacat.html#K3" target="_self">`11</a> </span>)</span> <a href="domain_1.html#K14">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="commacat.html#K4" target="_self">`12</a> </span>)</span> <a href="domain_1.html#K14">`11</a>  &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="mcart_1.html#K17">`11</a>  &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="commacat.html#K3" target="_self">`11</a> </span>)</span> <a href="domain_1.html#K14">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="commacat.html#K4" target="_self">`12</a> </span>)</span> <a href="domain_1.html#K14">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mcart_1.html#K18">`12</a>  &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="commacat.html#K3" target="_self">`11</a> </span>)</span> <a href="domain_1.html#K15">`12</a>  &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="commacat.html#K4" target="_self">`12</a> </span>)</span> <a href="domain_1.html#K15">`12</a>  &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="commacat.html#K3" target="_self">`11</a> </span>)</span> <a href="domain_1.html#K15">`12</a>  &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="commacat.html#K4" target="_self">`12</a> </span>)</span> <a href="domain_1.html#K15">`12</a>  &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E4:9_1_1_5"/><b>then </b>
( <font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">gf</font> <a href="commacat.html#K3" target="_self">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="mcart_1.html#K18">`12</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h1</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">gf</font> <a href="domain_1.html#K16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h1</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">gf</font> <a href="domain_1.html#K17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K17">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">hg</font> <a href="commacat.html#K4" target="_self">`12</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">hg</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="domain_1.html#K16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">hg</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="domain_1.html#K17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="mcart_1.html#K17">`11</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mcart_1.html#K17">`11</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font></span>)</span> <a href="mcart_1.html#K18">`12</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="mcart_1.html#K18">`12</a>  &amp; <font color="Maroon">hg</font> <a href="domain_1.html#K16">`21</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="domain_1.html#K16">`21</a> </span>)</span> &amp; <font color="Maroon">gf</font> <a href="domain_1.html#K16">`21</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="domain_1.html#K16">`21</a> </span>)</span> &amp; <font color="Maroon">hg</font> <a href="domain_1.html#K17">`22</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="domain_1.html#K17">`22</a> </span>)</span> &amp; <font color="Maroon">gf</font> <a href="domain_1.html#K17">`22</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="domain_1.html#K17">`22</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="domain_1.html#K16">`21</a> </span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="domain_1.html#K16">`21</a> </span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="domain_1.html#K17">`22</a> </span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="domain_1.html#K17">`22</a> </span>)</span></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="cat_1.html#T43">CAT_1:43</a></i>;<br/>
<b>hence </b><a NAME="E5:9_1_1_5"/>
<font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E19:9_1_1"/>
for <font color="Olive">b</font> being  <a href="cat_1.html#NM1">Object</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #) holds <br/> (  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; ( for <font color="Olive">f</font> being  <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #)  st  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Olive">b</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> ) &amp; ( for <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #)  st  <a href="cat_1.html#K2">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> holds <br/><font color="Olive">g</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">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">b</font> be   <a href="cat_1.html#NM1">Object</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #);<br/>

<b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> ;<br/>
<b>reconsider </b><font color="Maroon">i</font> =  <a href="cat_1.html#K5">id</a> <font color="Maroon">b</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> ;<br/>
<a NAME="E3:9_1_1_6"/>
<font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b'</font>,<font color="Maroon">b'</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><font color="Maroon">b'</font> <a href="domain_1.html#K14">`11</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><font color="Maroon">b'</font> <a href="domain_1.html#K15">`12</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:9_1_1_6"/><b>then </b><i><font color="Green">E84</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font></span>)</span> <a href="mcart_1.html#K17">`11</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font></span>)</span> <a href="mcart_1.html#K18">`12</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font></span>)</span> <a href="mcart_1.html#K19">`21</a>  <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="domain_1.html#K14">`11</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font></span>)</span> <a href="mcart_1.html#K20">`22</a>  <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="domain_1.html#K15">`12</a> </span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:9_1_1_6"/>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E6:9_1_1_6"/>
for <font color="Olive">f</font> being  <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #)  st  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">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">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #);<br/>

<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> ;<br/>
<b>assume </b><a NAME="E2:9_1_1_6_1"/>
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E3:9_1_1_6_1"/><b>then </b><i><font color="Green">E86</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Maroon">b</font></span>)</span> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="domain_1.html#K15">`12</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> <a href="domain_1.html#K15">`12</a>  &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="domain_1.html#K14">`11</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> <a href="domain_1.html#K14">`11</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mcart_1.html#K18">`12</a>  &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="commacat.html#K4" target="_self">`12</a> </span>)</span> <a href="domain_1.html#K14">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="commacat.html#K4" target="_self">`12</a> </span>)</span> <a href="domain_1.html#K15">`12</a>  )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E4:9_1_1_6_1"/><b>then </b><i><font color="Green">E87</font></i>: 
( <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="domain_1.html#K14">`11</a> </span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="domain_1.html#K16">`21</a>  &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="domain_1.html#K15">`12</a> </span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="domain_1.html#K17">`22</a>  )
 
<b>by </b><i><a class="ref" href="cat_1.html#T46">CAT_1:46</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f'</font> <a href="commacat.html#K3" target="_self">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="commacat.html#K4" target="_self">`12</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="domain_1.html#K16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">f'</font> <a href="domain_1.html#K17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i>, , , , </i>
;<br/>


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

<b>let </b><font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #);<br/>

<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> ;<br/>
<b>assume </b><a NAME="E8:9_1_1_6"/>
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E9:9_1_1_6"/><b>then </b><i><font color="Green">E89</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Maroon">b</font></span>)</span> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="domain_1.html#K15">`12</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> <a href="domain_1.html#K15">`12</a>  &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="domain_1.html#K14">`11</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> <a href="domain_1.html#K14">`11</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="mcart_1.html#K17">`11</a>  &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="commacat.html#K3" target="_self">`11</a> </span>)</span> <a href="domain_1.html#K14">`11</a>  &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="commacat.html#K3" target="_self">`11</a> </span>)</span> <a href="domain_1.html#K15">`12</a>  )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E10:9_1_1_6"/><b>then </b><i><font color="Green">E90</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="domain_1.html#K14">`11</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> <a href="domain_1.html#K16">`21</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="domain_1.html#K15">`12</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> <a href="domain_1.html#K17">`22</a>  )
 
<b>by </b><i><a class="ref" href="cat_1.html#T47">CAT_1:47</a></i>;<br/>
<b>thus </b><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="commacat.html#K3" target="_self">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">g'</font> <a href="commacat.html#K4" target="_self">`12</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g'</font> <a href="domain_1.html#K16">`21</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="domain_1.html#K16">`21</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g'</font> <a href="domain_1.html#K17">`22</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="domain_1.html#K17">`22</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">g</font>
<b>by </b><i>, , , , </i>
;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">FG</font> =  <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">M</font>,<font color="Maroon">D'</font>,<font color="Maroon">C'</font>,<font color="Maroon">'</font>,<font color="Maroon">I</font> #) as  <a href="cat_1.html#V1">strict</a> <a href="cat_1.html#NM4">Category</a> <b>by </b><i><a class="txt" href="commacat.html#E6"><i><font color="Green">Lemma17</font></i></a>, , , <a class="ref" href="cat_1.html#T29">CAT_1:29</a></i>;<br/>
<b>take </b>
<font color="Maroon">FG</font>
;<br/>

<b>thus </b><a NAME="E21:9_1_1"/>
( the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">FG</font> <a href="hidden.html#R1">=</a>  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> &amp; the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">FG</font> <a href="hidden.html#R1">=</a>  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> holds  the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> <a href="commacat.html#K3" target="_self">`11</a>  ) &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K2" target="_self">commaMorphs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> holds  the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> <a href="commacat.html#K4" target="_self">`12</a>  ) &amp; ( for <font color="Olive">o</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="commacat.html#K1" target="_self">commaObjs</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> holds  the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">o</font>,<font color="Olive">o</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><font color="Olive">o</font> <a href="domain_1.html#K14">`11</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><font color="Olive">o</font> <a href="domain_1.html#K15">`12</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> ) &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">FG</font> <a href="hidden.html#R1">=</a>  <a href="commacat.html#K6" target="_self">commaComp</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> )
 <b>by </b><i>, , </i>;<br/>


</div>
