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

<b>set </b><font color="Maroon" title="c1">M</font> = <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">a</font>,<font color="Olive" title="b2">b</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">a</font>, <font color="Olive" title="b2">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b3">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c2">a0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>consider </b><font color="Maroon" title="c3">f0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green" title="E7">A4</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">a0</font>,<font color="Maroon" title="c2">a0</font>,<font color="Maroon" title="c3">f0</font>] &amp; ( for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() holds <br/> ( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">a0</font>,<font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">g</font>] implies <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c3">f0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c2">a0</font>,<font color="Olive" title="b2">g</font>] implies <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">f0</font>,<font color="Olive" title="b2">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) ) ) )
 <b>by </b><i><a class="txt" href="cat_5.html#E2:8"><i><font color="Green" title="E5">A2</font></i></a></i>;<br/>
<a NAME="E4:8_1"/><i><font color="Green" title="E8">A5</font></i>: 
<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="c2">a0</font>,<font color="Maroon" title="c2">a0</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c3">f0</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> <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">a</font>,<font color="Olive" title="b2">b</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">a</font>, <font color="Olive" title="b2">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b3">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="cat_5.html#E3:8_1"><i><font color="Green" title="E7">A4</font></i></a></i>;<br/>
<a NAME="E5:8_1"/>
<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">a</font>,<font color="Olive" title="b2">b</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">a</font>, <font color="Olive" title="b2">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b3">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:8_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">a</font>,<font color="Olive" title="b2">b</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">a</font>, <font color="Olive" title="b2">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b3">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] </span>}</span> 
 ;<br/>

<a NAME="E2:8_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() ex <font color="Olive" title="b3">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon" title="c4">x</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">a</font>,<font color="Olive" title="b2">b</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> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] )
 
;<br/>
<b>hence </b><a NAME="E3:8_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">M</font> = <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">a</font>,<font color="Olive" title="b2">b</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">a</font>, <font color="Olive" title="b2">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b3">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <b>by </b><i><a class="txt" href="cat_5.html#E4:8_1"><i><font color="Green" title="E8">A5</font></i></a></i>;<br/>
<div><i><font color="Green" title="E9">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">M</font>;<br/><a NAME="E1:8_1_2"/>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c8">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E3:8_1_2"/><i><font color="Green" title="E10">A7</font></i>: 
( <font color="Maroon" title="c5">m</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="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c8">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">f</font>] )
 ;<br/><a NAME="E4:8_1_2"/>
( <font color="Maroon" title="c5">m</font> <a href="cat_5.html#K1" title="CAT_5:func.1">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">m</font> <a href="cat_5.html#K2" title="CAT_5:func.2">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c5">m</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E3:8_1_2"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</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="E5:8_1_2"/>
( <font color="Maroon" title="c5">m</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="c5">m</font> <a href="cat_5.html#K1" title="CAT_5:func.1">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="cat_5.html#K2" title="CAT_5:func.2">`12</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">m</font> <a href="cat_5.html#K1" title="CAT_5:func.1">`11</a> ,<font color="Maroon" title="c5">m</font> <a href="cat_5.html#K2" title="CAT_5:func.2">`12</a> ,<font color="Maroon" title="c5">m</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> ] )
 <b>by </b><i><a class="txt" href="cat_5.html#E3:8_1_2"><i><font color="Green" title="E10">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">M</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() = $1 <a href="cat_5.html#K1" title="CAT_5:func.1">`11</a> ;<br/>
<b>consider </b><font color="Maroon" title="c5">DM</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">M</font>,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E9:8_1"/><i><font color="Green" title="E10">A8</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">M</font> holds  <font color="Maroon" title="c5">DM</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">m</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>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">M</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() = $1 <a href="cat_5.html#K2" title="CAT_5:func.2">`12</a> ;<br/>
<b>consider </b><font color="Maroon" title="c6">CM</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">M</font>,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E11:8_1"/><i><font color="Green" title="E11">A9</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">M</font> holds  <font color="Maroon" title="c6">CM</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">m</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>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">$2 <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a> </span>)</span>,<span class="p3">(<span class="default">$1 <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>,<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default">$1 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>,<span class="p1">(<span class="default">$2 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> );<br/>
<div><i><font color="Green" title="E12">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:8_1_3"/><i><font color="Green" title="E13">A11</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>]
 ;<br/><b>then consider </b><font color="Maroon" title="c9">ax</font>, <font color="Maroon" title="c10">bx</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c11">fx</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E3:8_1_3"/><i><font color="Green" title="E14">A12</font></i>: 
( <font color="Maroon" title="c7">x</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="c9">ax</font>,<font color="Maroon" title="c10">bx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c11">fx</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">ax</font>,<font color="Maroon" title="c10">bx</font>,<font color="Maroon" title="c11">fx</font>] )
 ;<br/><b>consider </b><font color="Maroon" title="c12">ay</font>, <font color="Maroon" title="c13">b2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c14">fy</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E5:8_1_3"/><i><font color="Green" title="E15">A13</font></i>: 
( <font color="Maroon" title="c8">y</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="c12">ay</font>,<font color="Maroon" title="c13">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c14">fy</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c12">ay</font>,<font color="Maroon" title="c13">b2</font>,<font color="Maroon" title="c14">fy</font>] )
 <b>by </b><i><a class="txt" href="cat_5.html#E1:8_1_3"><i><font color="Green" title="E13">A11</font></i></a></i>;<br/><a NAME="E6:8_1_3"/><i><font color="Green" title="E16">A14</font></i>: 
( <font color="Maroon" title="c7">x</font> <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="c9">ax</font> &amp; <font color="Maroon" title="c7">x</font> <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="c10">bx</font> &amp; <font color="Maroon" title="c8">y</font> <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="c12">ay</font> &amp; <font color="Maroon" title="c8">y</font> <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="c13">b2</font> &amp; <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">fx</font> &amp; <font color="Maroon" title="c8">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">fy</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E3:8_1_3"><i><font color="Green" title="E14">A12</font></i></a>, <a class="txt" href="cat_5.html#E5:8_1_3"><i><font color="Green" title="E15">A13</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/><a NAME="E7:8_1_3"/><b>then </b>
( <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c11">fx</font>,<font color="Maroon" title="c14">fy</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c12">ay</font>,<font color="Maroon" title="c10">bx</font>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c11">fx</font>,<font color="Maroon" title="c14">fy</font>)] )
 <b>by </b><i><a class="txt" href="cat_5.html#E1:8"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="cat_5.html#E1:8_1_3"><i><font color="Green" title="E13">A11</font></i></a>, <a class="txt" href="cat_5.html#E3:8_1_3"><i><font color="Green" title="E14">A12</font></i></a>, <a class="txt" href="cat_5.html#E5:8_1_3"><i><font color="Green" title="E15">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E8:8_1_3"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="txt" href="cat_5.html#E6:8_1_3"><i><font color="Green" title="E16">A14</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c7">CC</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Maroon" title="c4">M</font><b> such that </b><br/><a NAME="E14:8_1"/><i><font color="Green" title="E13">A15</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">CC</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <b>and </b><br/><a NAME="E15:8_1"/><i><font color="Green" title="E14">A16</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">CC</font> holds <br/><font color="Maroon" title="c7">CC</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S6" title="BINOP_1:sch.6">BINOP_1:sch 6</a>(<a class="txt" href="cat_5.html#E12:8_1"><i><font color="Green" title="E12">A10</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(),   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( $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>,<font color="Olive" title="b1">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[$1,$1,<font color="Olive" title="b1">f</font>] &amp; ( for <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b3">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() holds <br/> ( ( <font color="Maroon">P<sub>1</sub></font>[$1,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">g</font>] implies <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b3">g</font>,<font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">b</font>,$1,<font color="Olive" title="b3">g</font>] implies <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) ) ) );<br/>
<div><i><font color="Green" title="E15">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>consider </b><font color="Maroon" title="c9">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E2:8_1_4"/><i><font color="Green" title="E16">A18</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">f</font>] &amp; ( for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() holds <br/> ( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">g</font>] implies <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c9">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c8">a</font>,<font color="Olive" title="b2">g</font>] implies <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c9">f</font>,<font color="Olive" title="b2">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) ) ) )
 <b>by </b><i><a class="txt" href="cat_5.html#E2:8"><i><font color="Green" title="E5">A2</font></i></a></i>;<br/><a NAME="E3:8_1_4"/>
<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="c8">a</font>,<font color="Maroon" title="c8">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c9">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> <font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="txt" href="cat_5.html#E2:8_1_4"><i><font color="Green" title="E16">A18</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">y</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"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c8">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c9">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">M</font> ;<br/><b>take </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">y</font>;<br/><b>thus </b><a NAME="E5:8_1_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c11">y</font>]
 <b>by </b><i><a class="txt" href="cat_5.html#E2:8_1_4"><i><font color="Green" title="E16">A18</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">I</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font><b> such that </b><br/><a NAME="E18:8_1"/><i><font color="Green" title="E16">A19</font></i>: 
for <font color="Olive" title="b1">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c8">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="cat_5.html#E16:8_1"><i><font color="Green" title="E15">A17</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c9">C</font> =  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #);<br/>
<a NAME="E19:8_1"/>
 <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) is <a href="cat_1.html#V2" title="CAT_1:attr.2">Category-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_5"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a>
<div class="add"><b>let </b><font color="Maroon" title="c10">f</font>, <font color="Maroon" title="c11">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">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #);<br/><a NAME="E1:8_1_5_1"/>
( ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">CC</font> implies ( <font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c11">g</font> <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="c10">f</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  &amp; <font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> ) ) &amp; ( <font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c11">g</font> <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="c10">f</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  &amp; <font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> &amp; <font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">CC</font> ) &amp; <font color="Maroon" title="c5">DM</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a>  &amp; <font color="Maroon" title="c6">CM</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  )
 <b>by </b><i><a class="txt" href="cat_5.html#E9:8_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="cat_5.html#E11:8_1"><i><font color="Green" title="E11">A9</font></i></a>, <a class="txt" href="cat_5.html#E14:8_1"><i><font color="Green" title="E13">A15</font></i></a></i>;<br/><b>hence </b><a NAME="E2:8_1_5_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) iff the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font> )
 ;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c10">f</font>, <font color="Maroon" title="c11">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">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #);<br/><a NAME="E1:8_1_5_2"/><i><font color="Green" title="E17">A20</font></i>: 
( the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a>  &amp; the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a>  &amp; the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  &amp; the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  )
 <b>by </b><i><a class="txt" href="cat_5.html#E9:8_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="cat_5.html#E11:8_1"><i><font color="Green" title="E11">A9</font></i></a></i>;<br/><b>assume </b><a NAME="E2:8_1_5_2"/>
the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font>
 ;<br/><a NAME="E3:8_1_5_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="c11">g</font>,<font color="Maroon" title="c10">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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">CC</font>
 <b>by </b><i><a class="txt" href="cat_5.html#E14:8_1"><i><font color="Green" title="E13">A15</font></i></a>, <a class="txt" href="cat_5.html#E1:8_1_5_2"><i><font color="Green" title="E17">A20</font></i></a></i>;<br/><a NAME="E4:8_1_5_2"/><b>then </b>
( <font color="Maroon" title="c7">CC</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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="c10">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="c11">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>,<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c7">CC</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">CC</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">CC</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">M</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E15:8_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:8_1_5_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></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="c10">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="c7">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></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="c11">g</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  &amp; <font color="Maroon" title="c7">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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> <font color="Maroon" title="c4">M</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="E6:8_1_5_2"/>
( the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font> &amp; the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E9:8_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="cat_5.html#E11:8_1"><i><font color="Green" title="E11">A9</font></i></a>, <a class="txt" href="cat_5.html#E1:8_1_5_2"><i><font color="Green" title="E17">A20</font></i></a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c10">f</font>, <font color="Maroon" title="c11">g</font>, <font color="Maroon" title="c12">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">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #);<br/><a NAME="E1:8_1_5_3"/><i><font color="Green" title="E17">A21</font></i>: 
( the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a>  &amp; the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a>  &amp; the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</font> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a>  &amp; the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  &amp; the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  &amp; the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  )
 <b>by </b><i><a class="txt" href="cat_5.html#E9:8_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="cat_5.html#E11:8_1"><i><font color="Green" title="E11">A9</font></i></a></i>;<br/><b>assume </b><a NAME="E2:8_1_5_3"/><i><font color="Green" title="E18">A22</font></i>: 
( the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> &amp; the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">f</font> )
 ;<br/><a NAME="E3:8_1_5_3"/><b>then </b><i><font color="Green" title="E19">A23</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">CC</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">h</font>,<font color="Maroon" title="c11">g</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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">CC</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E14:8_1"><i><font color="Green" title="E13">A15</font></i></a>, <a class="txt" href="cat_5.html#E1:8_1_5_3"><i><font color="Green" title="E17">A21</font></i></a></i>;<br/><a NAME="E4:8_1_5_3"/><b>then </b>
( <font color="Maroon" title="c7">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">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="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">CC</font> &amp; <font color="Maroon" title="c7">CC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">h</font>,<font color="Maroon" title="c11">g</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="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">CC</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">CC</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">M</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">gf</font> = <font color="Maroon" title="c7">CC</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font>, <font color="Maroon" title="c14">hg</font> = <font color="Maroon" title="c7">CC</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">h</font>,<font color="Maroon" title="c11">g</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">M</font> ;<br/><a NAME="E6:8_1_5_3"/><i><font color="Green" title="E20">A24</font></i>: 
( <font color="Maroon" title="c13">gf</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="c10">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="c11">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>,<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">hg</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="c11">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="c12">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>,<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="cat_5.html#E15:8_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="cat_5.html#E3:8_1_5_3"><i><font color="Green" title="E19">A23</font></i></a></i>;<br/><a NAME="E7:8_1_5_3"/><i><font color="Green" title="E21">A25</font></i>: 
( <font color="Maroon" title="c5">DM</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">gf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">gf</font> <a href="cat_5.html#K1" title="CAT_5:func.1">`11</a>  &amp; <font color="Maroon" title="c5">DM</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">hg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">hg</font> <a href="cat_5.html#K1" title="CAT_5:func.1">`11</a>  &amp; <font color="Maroon" title="c6">CM</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">gf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">gf</font> <a href="cat_5.html#K2" title="CAT_5:func.2">`12</a>  &amp; <font color="Maroon" title="c6">CM</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">hg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">hg</font> <a href="cat_5.html#K2" title="CAT_5:func.2">`12</a>  )
 <b>by </b><i><a class="txt" href="cat_5.html#E9:8_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="cat_5.html#E11:8_1"><i><font color="Green" title="E11">A9</font></i></a></i>;<br/><a NAME="E8:8_1_5_3"/><b>then </b><i><font color="Green" title="E22">A26</font></i>: 
( <font color="Maroon" title="c5">DM</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">gf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a>  &amp; <font color="Maroon" title="c5">DM</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">hg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a>  &amp; <font color="Maroon" title="c6">CM</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">gf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  &amp; <font color="Maroon" title="c6">CM</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">hg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a>  )
 <b>by </b><i><a class="txt" href="cat_5.html#E6:8_1_5_3"><i><font color="Green" title="E20">A24</font></i></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/><a NAME="E9:8_1_5_3"/><b>then </b><i><font color="Green" title="E23">A27</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">h</font>,<font color="Maroon" title="c13">gf</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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">CC</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">hg</font>,<font color="Maroon" title="c10">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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">CC</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E14:8_1"><i><font color="Green" title="E13">A15</font></i></a>, <a class="txt" href="cat_5.html#E1:8_1_5_3"><i><font color="Green" title="E17">A21</font></i></a>, <a class="txt" href="cat_5.html#E2:8_1_5_3"><i><font color="Green" title="E18">A22</font></i></a>, <a class="txt" href="cat_5.html#E7:8_1_5_3"><i><font color="Green" title="E21">A25</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">f'</font> = <font color="Maroon" title="c10">f</font>, <font color="Maroon" title="c16">g'</font> = <font color="Maroon" title="c11">g</font>, <font color="Maroon" title="c17">h'</font> = <font color="Maroon" title="c12">h</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">M</font> ;<br/><a NAME="E11:8_1_5_3"/><i><font color="Green" title="E24">A28</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c15">f'</font> <a href="cat_5.html#K1" title="CAT_5:func.1">`11</a> ,<font color="Maroon" title="c15">f'</font> <a href="cat_5.html#K2" title="CAT_5:func.2">`12</a> ,<font color="Maroon" title="c15">f'</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> ] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c16">g'</font> <a href="cat_5.html#K1" title="CAT_5:func.1">`11</a> ,<font color="Maroon" title="c16">g'</font> <a href="cat_5.html#K2" title="CAT_5:func.2">`12</a> ,<font color="Maroon" title="c16">g'</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> ] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c17">h'</font> <a href="cat_5.html#K1" title="CAT_5:func.1">`11</a> ,<font color="Maroon" title="c17">h'</font> <a href="cat_5.html#K2" title="CAT_5:func.2">`12</a> ,<font color="Maroon" title="c17">h'</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> ] )
 <b>by </b><i><a class="txt" href="cat_5.html#E7:8_1"><i><font color="Green" title="E9">A6</font></i></a></i>;<br/><b>thus </b>the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">h</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<font color="Maroon" title="c10">f</font></span>)</span> = 
<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="c10">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="c12">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>,<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">gf</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> </span>)</span>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E15:8_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="cat_5.html#E7:8_1_5_3"><i><font color="Green" title="E21">A25</font></i></a>, <a class="txt" href="cat_5.html#E8:8_1_5_3"><i><font color="Green" title="E22">A26</font></i></a>, <a class="txt" href="cat_5.html#E9:8_1_5_3"><i><font color="Green" title="E23">A27</font></i></a></i>
<br/>.= 
<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="c10">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="c12">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>,<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c17">h'</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> </span>)</span>,<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c16">g'</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">f'</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> </span>)</span>))</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E6:8_1_5_3"><i><font color="Green" title="E20">A24</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<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="c10">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="c12">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>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c17">h'</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">g'</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> </span>)</span>),<span class="p1">(<span class="default"><font color="Maroon" title="c15">f'</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> </span>)</span>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E3:8"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="cat_5.html#E1:8_1_5_3"><i><font color="Green" title="E17">A21</font></i></a>, <a class="txt" href="cat_5.html#E2:8_1_5_3"><i><font color="Green" title="E18">A22</font></i></a>, <a class="txt" href="cat_5.html#E11:8_1_5_3"><i><font color="Green" title="E24">A28</font></i></a></i>
<br/>.= 
<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="c10">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="c12">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>,<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c14">hg</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E6:8_1_5_3"><i><font color="Green" title="E20">A24</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">h</font>,<font color="Maroon" title="c11">g</font></span>)</span>,<font color="Maroon" title="c10">f</font>
<b>by </b><i><a class="txt" href="cat_5.html#E15:8_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="cat_5.html#E7:8_1_5_3"><i><font color="Green" title="E21">A25</font></i></a>, <a class="txt" href="cat_5.html#E8:8_1_5_3"><i><font color="Green" title="E22">A26</font></i></a>, <a class="txt" href="cat_5.html#E9:8_1_5_3"><i><font color="Green" title="E23">A27</font></i></a></i>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c10">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">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #);<br/>

<b>consider </b><font color="Maroon" title="c11">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E5:8_1_5"/><i><font color="Green" title="E17">A29</font></i>: 
( <font color="Maroon" title="c8">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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="c10">b</font>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c11">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c11">f</font>] &amp; ( for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() holds <br/> ( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c10">b</font>,<font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">g</font>] implies <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c11">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c10">b</font>,<font color="Olive" title="b2">g</font>] implies <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c11">f</font>,<font color="Olive" title="b2">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) ) ) )
 <b>by </b><i><a class="txt" href="cat_5.html#E18:8_1"><i><font color="Green" title="E16">A19</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">b'</font> = <font color="Maroon" title="c10">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">Ib</font> = <font color="Maroon" title="c8">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">M</font> ;<br/>
<b>thus </b>the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a> 
<b>by </b><i><a class="txt" href="cat_5.html#E9:8_1"><i><font color="Green" title="E10">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="cat_5.html#E5:8_1_5"><i><font color="Green" title="E17">A29</font></i></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>
;<br/>

<b>thus </b>the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a> 
<b>by </b><i><a class="txt" href="cat_5.html#E11:8_1"><i><font color="Green" title="E11">A9</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="cat_5.html#E5:8_1_5"><i><font color="Green" title="E17">A29</font></i></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>
;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c14">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">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #);<br/><b>reconsider </b><font color="Maroon" title="c15">g</font> = <font color="Maroon" title="c14">f'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">M</font> ;<br/><b>assume </b><a NAME="E2:8_1_5_6"/>
the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 ;<br/><a NAME="E3:8_1_5_6"/><b>then </b><i><font color="Green" title="E18">A30</font></i>: 
( <font color="Maroon" title="c15">g</font> <a href="cat_5.html#K2" title="CAT_5:func.2">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> &amp; <font color="Maroon" title="c13">Ib</font> <a href="cat_5.html#K1" title="CAT_5:func.1">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E11:8_1"><i><font color="Green" title="E11">A9</font></i></a>, <a class="txt" href="cat_5.html#E5:8_1_5"><i><font color="Green" title="E17">A29</font></i></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/><a NAME="E4:8_1_5_6"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c15">g</font> <a href="cat_5.html#K1" title="CAT_5:func.1">`11</a> ,<font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c15">g</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> ] &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">Ib</font>,<font color="Maroon" title="c15">g</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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">CC</font> &amp; <font color="Maroon" title="c13">Ib</font> <a href="cat_5.html#K2" title="CAT_5:func.2">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> &amp; <font color="Maroon" title="c13">Ib</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E7:8_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="cat_5.html#E14:8_1"><i><font color="Green" title="E13">A15</font></i></a>, <a class="txt" href="cat_5.html#E5:8_1_5"><i><font color="Green" title="E17">A29</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/><a NAME="E5:8_1_5_6"/><b>then </b>
( <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> </span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a>  &amp; <font color="Maroon" title="c7">CC</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">Ib</font>,<font color="Maroon" title="c15">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="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="cat_5.html#K1" title="CAT_5:func.1">`11</a> </span>)</span>,<font color="Maroon" title="c10">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> </span>)</span>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="cat_5.html#E15:8_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="cat_5.html#E5:8_1_5"><i><font color="Green" title="E17">A29</font></i></a></i>;<br/><b>hence </b><a NAME="E6:8_1_5_6"/>
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span>,<font color="Maroon" title="c14">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f'</font>
 <b>by </b><i><a class="txt" href="cat_5.html#E7:8_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="cat_5.html#E3:8_1_5_6"><i><font color="Green" title="E18">A30</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c14">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">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c15">g</font> = <font color="Maroon" title="c14">f'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">M</font> ;<br/>
<b>assume </b><a NAME="E12:8_1_5"/>
the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 ;<br/>

<a NAME="E13:8_1_5"/><b>then </b><i><font color="Green" title="E18">A31</font></i>: 
( <font color="Maroon" title="c15">g</font> <a href="cat_5.html#K1" title="CAT_5:func.1">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> &amp; <font color="Maroon" title="c13">Ib</font> <a href="cat_5.html#K2" title="CAT_5:func.2">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E9:8_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="cat_5.html#E5:8_1_5"><i><font color="Green" title="E17">A29</font></i></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/>
<a NAME="E14:8_1_5"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c15">g</font> <a href="cat_5.html#K2" title="CAT_5:func.2">`12</a> ,<font color="Maroon" title="c15">g</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> ] &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c13">Ib</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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">CC</font> &amp; <font color="Maroon" title="c13">Ib</font> <a href="cat_5.html#K1" title="CAT_5:func.1">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> &amp; <font color="Maroon" title="c13">Ib</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E7:8_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="cat_5.html#E14:8_1"><i><font color="Green" title="E13">A15</font></i></a>, <a class="txt" href="cat_5.html#E5:8_1_5"><i><font color="Green" title="E17">A29</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/>
<a NAME="E15:8_1_5"/><b>then </b>
( <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> </span>)</span>,<font color="Maroon" title="c11">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a>  &amp; <font color="Maroon" title="c7">CC</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">g</font>,<font color="Maroon" title="c13">Ib</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="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">b</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="cat_5.html#K2" title="CAT_5:func.2">`12</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> </span>)</span>,<font color="Maroon" title="c11">f</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E15:8_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="cat_5.html#E5:8_1_5"><i><font color="Green" title="E17">A29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:8_1_5"/>
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">f'</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">I</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f'</font>
 <b>by </b><i><a class="txt" href="cat_5.html#E7:8_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="cat_5.html#E13:8_1_5"><i><font color="Green" title="E18">A31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">C</font> =  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">DM</font>,<font color="Maroon" title="c6">CM</font>,<font color="Maroon" title="c7">CC</font>,<font color="Maroon" title="c8">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> ;<br/>
<a NAME="E21:8_1"/>
<font color="Maroon" title="c10">C</font> is <a href="cat_5.html#V1" title="CAT_5:attr.1">with_triple-like_morphisms</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_5.html#D1" target="_self" title="CAT_5:def.1">CAT_5:def 1</a><br/>

<a NAME="E1:8_1_6"/>
<font color="Maroon" title="c11">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c14">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E3:8_1_6"/><i><font color="Green" title="E17">A32</font></i>: 
( <font color="Maroon" title="c11">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"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c14">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">g</font>] )
 ;<br/>
<b>take </b>
<font color="Maroon" title="c14">g</font>
;<br/>

<i><font color="Green" title="E18">A33</font></i>:  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c11">f</font> = 
<font color="Maroon" title="c11">f</font> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a> 
<b>by </b><i><a class="txt" href="cat_5.html#E9:8_1"><i><font color="Green" title="E10">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c12">a</font>
<b>by </b><i><a class="txt" href="cat_5.html#E3:8_1_6"><i><font color="Green" title="E17">A32</font></i></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>
;<br/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c11">f</font> = 
<font color="Maroon" title="c11">f</font> <a href="mcart_1.html#K18" title="MCART_1:func.18">`12</a> 
<b>by </b><i><a class="txt" href="cat_5.html#E11:8_1"><i><font color="Green" title="E11">A9</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">b</font>
<b>by </b><i><a class="txt" href="cat_5.html#E3:8_1_6"><i><font color="Green" title="E17">A32</font></i></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="E6:8_1_6"/>
<font color="Maroon" title="c11">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="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c11">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c11">f</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c14">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="cat_5.html#E3:8_1_6"><i><font color="Green" title="E17">A32</font></i></a>, <a class="txt" href="cat_5.html#E4:8_1_6"><i><font color="Green" title="E18">A33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c11">C</font> = <font color="Maroon" title="c10">C</font> as  <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a> <a href="cat_5.html#V1" title="CAT_5:attr.1">with_triple-like_morphisms</a> <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c11">C</font>
;<br/>

<b>thus </b><a NAME="E23:8_1"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c14">f</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>();<br/><b>assume </b><a NAME="E1:8_1_7"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">f</font>]
 ;<br/><a NAME="E2:8_1_7"/><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="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c14">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> <font color="Maroon" title="c4">M</font>
 ;<br/><b>hence </b><a NAME="E3:8_1_7"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c14">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">C</font>
 ;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c12">m</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">C</font>;<br/><a NAME="E1:8_1_8"/>
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font>
 ;<br/><b>hence </b><a NAME="E2:8_1_8"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() ex <font color="Olive" title="b3">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon" title="c12">m</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">a</font>,<font color="Olive" title="b2">b</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> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] )
 ;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c12">m1</font>, <font color="Maroon" title="c13">m2</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">C</font>;<br/><b>let </b><font color="Maroon" title="c14">a1</font>, <font color="Maroon" title="c15">a2</font>, <font color="Maroon" title="c16">a3</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c17">f1</font>, <font color="Maroon" title="c18">f2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>





<b>assume </b><a NAME="E26:8_1"/><i><font color="Green" title="E17">A34</font></i>: 
( <font color="Maroon" title="c12">m1</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="c14">a1</font>,<font color="Maroon" title="c15">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c17">f1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c13">m2</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="c15">a2</font>,<font color="Maroon" title="c16">a3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c18">f2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 ;<br/>

<a NAME="E27:8_1"/><b>then </b><i><font color="Green" title="E18">A35</font></i>: 
( <font color="Maroon" title="c12">m1</font> <a href="cat_5.html#K4" title="CAT_5:func.4">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a1</font> &amp; <font color="Maroon" title="c12">m1</font> <a href="cat_5.html#K5" title="CAT_5:func.5">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a2</font> &amp; <font color="Maroon" title="c13">m2</font> <a href="cat_5.html#K4" title="CAT_5:func.4">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a2</font> &amp; <font color="Maroon" title="c13">m2</font> <a href="cat_5.html#K5" title="CAT_5:func.5">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a3</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/>
<a NAME="E28:8_1"/><b>then </b><i><font color="Green" title="E19">A36</font></i>: 
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c13">m2</font>,<font color="Maroon" title="c12">m1</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">CC</font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E14:8_1"><i><font color="Green" title="E13">A15</font></i></a></i>;<br/>
<b>hence </b><font color="Maroon" title="c13">m2</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c12">m1</font> = 
<font color="Maroon" title="c7">CC</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">m2</font>,<font color="Maroon" title="c12">m1</font>
<b>by </b><i><a class="ref" href="cat_1.html#D4" title="CAT_1:def.4">CAT_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</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="c14">a1</font>,<font color="Maroon" title="c16">a3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c13">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E15:8_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="cat_5.html#E27:8_1"><i><font color="Green" title="E18">A35</font></i></a>, <a class="txt" href="cat_5.html#E28:8_1"><i><font color="Green" title="E19">A36</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</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="c14">a1</font>,<font color="Maroon" title="c16">a3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c18">f2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E26:8_1"><i><font color="Green" title="E17">A34</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</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="c14">a1</font>,<font color="Maroon" title="c16">a3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c18">f2</font>,<font color="Maroon" title="c17">f1</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E26:8_1"><i><font color="Green" title="E17">A34</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/><br/>


</div>
