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

<span class="kw">set </span><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/>
<span class="kw">consider </span><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/>
<span class="kw">consider </span><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>()<span class="kw"> such that </span><br/><a NAME="E3:8_1"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<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>]
 <span class="kw">and </span><br/><a NAME="E4:8_1"/>
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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E5">A2</font></span></a></span>;<br/>
<a NAME="E5:8_1"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a></span>;<br/>
<a NAME="E6:8_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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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>  or <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> )</span><br/>

<span class="kw">assume </span><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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">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></span><br/>

<a NAME="E2:8_1_1"/><span class="kw">then </span>
 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/>
<span class="kw">hence </span><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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E9">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> ] )</span><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/><span class="kw">then </span><span class="kw">consider </span><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>()<span class="kw"> such that </span><br/><a NAME="E3:8_1_2"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:8_1_2"/><span class="lab"><font color="Green" title="E11">A8</font></span>: 
<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="E5:8_1_2"/><span class="lab"><font color="Green" title="E12">A9</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_2"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/><a NAME="E6:8_1_2"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_2"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/><span class="kw">hence </span><a NAME="E7:8_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> ] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_2"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E4:8_1_2"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="txt" href="#E5:8_1_2"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">deffunc </span><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>) <span class="kw">-&gt; </span>   <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/>
<span class="kw">consider </span><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>()<span class="kw"> such that </span><br/><a NAME="E10:8_1"/><span class="lab"><font color="Green" title="E10">A10</font></span>: 
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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span>
<span class="kw">deffunc </span><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>) <span class="kw">-&gt; </span>   <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/>
<span class="kw">consider </span><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>()<span class="kw"> such that </span><br/><a NAME="E12:8_1"/><span class="lab"><font color="Green" title="E11">A11</font></span>: 
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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span>
<span class="kw">deffunc </span><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> ) <span class="kw">-&gt; </span>   <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/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span>( $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><span class="lab"><font color="Green" title="E12">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>] implies <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> )</span><br/><span class="kw">assume </span><a NAME="E1:8_1_3"/><span class="lab"><font color="Green" title="E13">A13</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">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></span><br/><span class="kw">then </span><span class="kw">consider </span><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>()<span class="kw"> such that </span><br/><a NAME="E3:8_1_3"/><span class="lab"><font color="Green" title="E14">A14</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:8_1_3"/><span class="lab"><font color="Green" title="E15">A15</font></span>: 
<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/><span class="kw">consider </span><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>()<span class="kw"> such that </span><br/><a NAME="E6:8_1_3"/><span class="lab"><font color="Green" title="E16">A16</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:8_1_3"/><span class="lab"><font color="Green" title="E17">A17</font></span>: 
<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>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1_3"><span class="lab"><font color="Green" title="E13">A13</font></span></a></span>;<br/><a NAME="E8:8_1_3"/><span class="lab"><font color="Green" title="E18">A18</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_3"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/><a NAME="E9:8_1_3"/><span class="lab"><font color="Green" title="E19">A19</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_3"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/><a NAME="E10:8_1_3"/><span class="lab"><font color="Green" title="E20">A20</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1_3"><span class="lab"><font color="Green" title="E16">A16</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/><a NAME="E11:8_1_3"/><span class="lab"><font color="Green" title="E21">A21</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1_3"><span class="lab"><font color="Green" title="E16">A16</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/><a NAME="E12:8_1_3"/><span class="lab"><font color="Green" title="E22">A22</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_3"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/><a NAME="E13:8_1_3"/><span class="lab"><font color="Green" title="E23">A23</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1_3"><span class="lab"><font color="Green" title="E16">A16</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/><a NAME="E14:8_1_3"/><span class="lab"><font color="Green" title="E24">A24</font></span>: 
<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>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E1:8_1_3"><span class="lab"><font color="Green" title="E13">A13</font></span></a>, <a class="txt" href="#E4:8_1_3"><span class="lab"><font color="Green" title="E15">A15</font></span></a>, <a class="txt" href="#E7:8_1_3"><span class="lab"><font color="Green" title="E17">A17</font></span></a>, <a class="txt" href="#E8:8_1_3"><span class="lab"><font color="Green" title="E18">A18</font></span></a>, <a class="txt" href="#E11:8_1_3"><span class="lab"><font color="Green" title="E21">A21</font></span></a></span>;<br/><a NAME="E15:8_1_3"/>
<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>)]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E1:8_1_3"><span class="lab"><font color="Green" title="E13">A13</font></span></a>, <a class="txt" href="#E4:8_1_3"><span class="lab"><font color="Green" title="E15">A15</font></span></a>, <a class="txt" href="#E7:8_1_3"><span class="lab"><font color="Green" title="E17">A17</font></span></a>, <a class="txt" href="#E8:8_1_3"><span class="lab"><font color="Green" title="E18">A18</font></span></a>, <a class="txt" href="#E11:8_1_3"><span class="lab"><font color="Green" title="E21">A21</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E16:8_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:8_1_3"><span class="lab"><font color="Green" title="E19">A19</font></span></a>, <a class="txt" href="#E10:8_1_3"><span class="lab"><font color="Green" title="E20">A20</font></span></a>, <a class="txt" href="#E12:8_1_3"><span class="lab"><font color="Green" title="E22">A22</font></span></a>, <a class="txt" href="#E13:8_1_3"><span class="lab"><font color="Green" title="E23">A23</font></span></a>, <a class="txt" href="#E14:8_1_3"><span class="lab"><font color="Green" title="E24">A24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E15:8_1"/><span class="lab"><font color="Green" title="E13">A25</font></span>: 
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#K1" title="RELSET_1:func.1">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>] ) )
 <span class="kw">and </span><br/><a NAME="E16:8_1"/><span class="lab"><font color="Green" title="E14">A26</font></span>: 
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#K1" title="RELSET_1:func.1">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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S6" title="BINOP_1:sch.6">BINOP_1:sch 6</a>(<a class="txt" href="#E13:8_1"><span class="lab"><font color="Green" title="E12">A12</font></span></a>);<br/></span>
<span class="kw">defpred </span><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> ] <span class="kw">means </span> 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><span class="lab"><font color="Green" title="E15">A27</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">M</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Olive" title="b1">y</font>]</span><br/><span class="kw">consider </span><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>()<span class="kw"> such that </span><br/><a NAME="E2:8_1_4"/><span class="lab"><font color="Green" title="E16">A28</font></span>: 
<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>]
 <span class="kw">and </span><br/><a NAME="E3:8_1_4"/><span class="lab"><font color="Green" title="E17">A29</font></span>: 
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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E5">A2</font></span></a></span>;<br/><a NAME="E4:8_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_4"><span class="lab"><font color="Green" title="E16">A28</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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/><span class="kw">take </span><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c11">y</font>]</span><br/><span class="kw">thus </span><a NAME="E6:8_1_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c11">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_4"><span class="lab"><font color="Green" title="E16">A28</font></span></a>, <a class="txt" href="#E3:8_1_4"><span class="lab"><font color="Green" title="E17">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E19:8_1"/><span class="lab"><font color="Green" title="E16">A30</font></span>: 
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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E17:8_1"><span class="lab"><font color="Green" title="E15">A27</font></span></a>);<br/></span>
<span class="kw">set </span><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="E20:8_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"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</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> #)  holds <br/>(  not the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> <font color="Olive">b<sub>1</sub></font> or ( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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> <font color="Olive">b<sub>1</sub></font> &amp; the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> <font color="Olive">b<sub>2</sub></font> ) ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</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> #)  holds <br/>(  not the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> <font color="Olive">b<sub>2</sub></font> or  not the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> <font color="Olive">b<sub>1</sub></font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="Olive">b<sub>3</sub></font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> #) holds <br/> ( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</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> #)  holds <br/>(  not the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</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> #)  holds <br/>(  not the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="Olive">b<sub>2</sub></font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) ) ) )</span>
<div class="add"><span class="kw">let </span><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> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="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#K1" title="RELSET_1:func.1">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> )</span><br/><a NAME="E1:8_1_5_1"/><span class="lab"><font color="Green" title="E17">A31</font></span>: 
( <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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">CC</font> iff ( <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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8_1"><span class="lab"><font color="Green" title="E13">A25</font></span></a></span>;<br/><a NAME="E2:8_1_5_1"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:8_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#K1" title="RELSET_1:func.1">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:8_1"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="txt" href="#E1:8_1_5_1"><span class="lab"><font color="Green" title="E17">A31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</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> #)  holds <br/>(  not the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> <font color="Olive">b<sub>2</sub></font> or  not the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> <font color="Olive">b<sub>1</sub></font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="Olive">b<sub>3</sub></font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> #) holds <br/> ( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</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> #)  holds <br/>(  not the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</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> #)  holds <br/>(  not the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="Olive">b<sub>2</sub></font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) ) ) )</span>
<div class="add"><span class="kw">let </span><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> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> implies ( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> ) )</span><br/><a NAME="E1:8_1_5_2"/><span class="lab"><font color="Green" title="E17">A32</font></span>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a></span>;<br/><a NAME="E2:8_1_5_2"/><span class="lab"><font color="Green" title="E18">A33</font></span>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a></span>;<br/><a NAME="E3:8_1_5_2"/><span class="lab"><font color="Green" title="E19">A34</font></span>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:8_1"><span class="lab"><font color="Green" title="E11">A11</font></span></a></span>;<br/><a NAME="E4:8_1_5_2"/><span class="lab"><font color="Green" title="E20">A35</font></span>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:8_1"><span class="lab"><font color="Green" title="E11">A11</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E5:8_1_5_2"/>
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> )</span><br/><a NAME="E6:8_1_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A36</font></span>: 
<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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">CC</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8_1"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E2:8_1_5_2"><span class="lab"><font color="Green" title="E18">A33</font></span></a>, <a class="txt" href="#E3:8_1_5_2"><span class="lab"><font color="Green" title="E19">A34</font></span></a></span>;<br/><a NAME="E7:8_1_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A37</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:8_1"><span class="lab"><font color="Green" title="E14">A26</font></span></a></span>;<br/><a NAME="E8:8_1_5_2"/><span class="lab"><font color="Green" title="E23">A38</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">CC</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1_5_2"><span class="lab"><font color="Green" title="E21">A36</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E9:8_1_5_2"/><span class="lab"><font color="Green" title="E24">A39</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:8_1_5_2"><span class="lab"><font color="Green" title="E22">A37</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/><a NAME="E10:8_1_5_2"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:8_1_5_2"><span class="lab"><font color="Green" title="E22">A37</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/><span class="kw">hence </span><a NAME="E11:8_1_5_2"/>
( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="txt" href="#E12:8_1"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="txt" href="#E1:8_1_5_2"><span class="lab"><font color="Green" title="E17">A32</font></span></a>, <a class="txt" href="#E4:8_1_5_2"><span class="lab"><font color="Green" title="E20">A35</font></span></a>, <a class="txt" href="#E8:8_1_5_2"><span class="lab"><font color="Green" title="E23">A38</font></span></a>, <a class="txt" href="#E9:8_1_5_2"><span class="lab"><font color="Green" title="E24">A39</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> #) holds <br/> ( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</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> #)  holds <br/>(  not the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</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> #)  holds <br/>(  not the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="Olive">b<sub>2</sub></font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) )</span>
<div class="add"><span class="kw">let </span><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> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> implies the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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> )</span><br/><a NAME="E1:8_1_5_3"/><span class="lab"><font color="Green" title="E17">A40</font></span>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a></span>;<br/><a NAME="E2:8_1_5_3"/><span class="lab"><font color="Green" title="E18">A41</font></span>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a></span>;<br/><a NAME="E3:8_1_5_3"/><span class="lab"><font color="Green" title="E19">A42</font></span>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:8_1"><span class="lab"><font color="Green" title="E11">A11</font></span></a></span>;<br/><a NAME="E4:8_1_5_3"/><span class="lab"><font color="Green" title="E20">A43</font></span>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:8_1"><span class="lab"><font color="Green" title="E11">A11</font></span></a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:8_1_5_3"/><span class="lab"><font color="Green" title="E21">A44</font></span>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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>
 <span class="kw">and </span><br/><a NAME="E6:8_1_5_3"/><span class="lab"><font color="Green" title="E22">A45</font></span>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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></span><br/><a NAME="E7:8_1_5_3"/><span class="lab"><font color="Green" title="E23">A46</font></span>: 
<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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">CC</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8_1"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E1:8_1_5_3"><span class="lab"><font color="Green" title="E17">A40</font></span></a>, <a class="txt" href="#E3:8_1_5_3"><span class="lab"><font color="Green" title="E19">A42</font></span></a>, <a class="txt" href="#E6:8_1_5_3"><span class="lab"><font color="Green" title="E22">A45</font></span></a></span>;<br/><a NAME="E8:8_1_5_3"/><span class="lab"><font color="Green" title="E24">A47</font></span>: 
<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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">CC</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8_1"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E2:8_1_5_3"><span class="lab"><font color="Green" title="E18">A41</font></span></a>, <a class="txt" href="#E4:8_1_5_3"><span class="lab"><font color="Green" title="E20">A43</font></span></a>, <a class="txt" href="#E5:8_1_5_3"><span class="lab"><font color="Green" title="E21">A44</font></span></a></span>;<br/><a NAME="E9:8_1_5_3"/><span class="lab"><font color="Green" title="E25">A48</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">CC</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:8_1_5_3"><span class="lab"><font color="Green" title="E23">A46</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E10:8_1_5_3"/>
<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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">CC</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8_1_5_3"><span class="lab"><font color="Green" title="E24">A47</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:8_1_5_3"><span class="lab"><font color="Green" title="E25">A48</font></span></a></span>;<br/><a NAME="E12:8_1_5_3"/><span class="lab"><font color="Green" title="E26">A49</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:8_1"><span class="lab"><font color="Green" title="E14">A26</font></span></a>, <a class="txt" href="#E7:8_1_5_3"><span class="lab"><font color="Green" title="E23">A46</font></span></a></span>;<br/><a NAME="E13:8_1_5_3"/><span class="lab"><font color="Green" title="E27">A50</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:8_1"><span class="lab"><font color="Green" title="E14">A26</font></span></a>, <a class="txt" href="#E8:8_1_5_3"><span class="lab"><font color="Green" title="E24">A47</font></span></a></span>;<br/><a NAME="E14:8_1_5_3"/><span class="lab"><font color="Green" title="E28">A51</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a></span>;<br/><a NAME="E15:8_1_5_3"/><span class="lab"><font color="Green" title="E29">A52</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a></span>;<br/><a NAME="E16:8_1_5_3"/><span class="lab"><font color="Green" title="E30">A53</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:8_1"><span class="lab"><font color="Green" title="E11">A11</font></span></a></span>;<br/><a NAME="E17:8_1_5_3"/><span class="lab"><font color="Green" title="E31">A54</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:8_1"><span class="lab"><font color="Green" title="E11">A11</font></span></a></span>;<br/><a NAME="E18:8_1_5_3"/><span class="lab"><font color="Green" title="E32">A55</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:8_1_5_3"><span class="lab"><font color="Green" title="E26">A49</font></span></a>, <a class="txt" href="#E14:8_1_5_3"><span class="lab"><font color="Green" title="E28">A51</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/><a NAME="E19:8_1_5_3"/><span class="lab"><font color="Green" title="E33">A56</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_5_3"><span class="lab"><font color="Green" title="E27">A50</font></span></a>, <a class="txt" href="#E15:8_1_5_3"><span class="lab"><font color="Green" title="E29">A52</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/><a NAME="E20:8_1_5_3"/><span class="lab"><font color="Green" title="E34">A57</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:8_1_5_3"><span class="lab"><font color="Green" title="E26">A49</font></span></a>, <a class="txt" href="#E16:8_1_5_3"><span class="lab"><font color="Green" title="E30">A53</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/><a NAME="E21:8_1_5_3"/><span class="lab"><font color="Green" title="E35">A58</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_5_3"><span class="lab"><font color="Green" title="E27">A50</font></span></a>, <a class="txt" href="#E17:8_1_5_3"><span class="lab"><font color="Green" title="E31">A54</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/><a NAME="E22:8_1_5_3"/><span class="lab"><font color="Green" title="E36">A59</font></span>: 
<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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">CC</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8_1"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E2:8_1_5_3"><span class="lab"><font color="Green" title="E18">A41</font></span></a>, <a class="txt" href="#E4:8_1_5_3"><span class="lab"><font color="Green" title="E20">A43</font></span></a>, <a class="txt" href="#E5:8_1_5_3"><span class="lab"><font color="Green" title="E21">A44</font></span></a>, <a class="txt" href="#E16:8_1_5_3"><span class="lab"><font color="Green" title="E30">A53</font></span></a>, <a class="txt" href="#E20:8_1_5_3"><span class="lab"><font color="Green" title="E34">A57</font></span></a></span>;<br/><a NAME="E23:8_1_5_3"/><span class="lab"><font color="Green" title="E37">A60</font></span>: 
<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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">CC</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8_1"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E1:8_1_5_3"><span class="lab"><font color="Green" title="E17">A40</font></span></a>, <a class="txt" href="#E3:8_1_5_3"><span class="lab"><font color="Green" title="E19">A42</font></span></a>, <a class="txt" href="#E6:8_1_5_3"><span class="lab"><font color="Green" title="E22">A45</font></span></a>, <a class="txt" href="#E15:8_1_5_3"><span class="lab"><font color="Green" title="E29">A52</font></span></a>, <a class="txt" href="#E19:8_1_5_3"><span class="lab"><font color="Green" title="E33">A56</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">f9</font> = <font color="Maroon" title="c10">f</font>, <font color="Maroon" title="c16">g9</font> = <font color="Maroon" title="c11">g</font>, <font color="Maroon" title="c17">h9</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="E25:8_1_5_3"/><span class="lab"><font color="Green" title="E38">A61</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c15">f9</font> <a href="cat_5.html#K1" title="CAT_5:func.1">`11</a> ,<font color="Maroon" title="c15">f9</font> <a href="cat_5.html#K2" title="CAT_5:func.2">`12</a> ,<font color="Maroon" title="c15">f9</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a></span>;<br/><a NAME="E26:8_1_5_3"/><span class="lab"><font color="Green" title="E39">A62</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c16">g9</font> <a href="cat_5.html#K1" title="CAT_5:func.1">`11</a> ,<font color="Maroon" title="c16">g9</font> <a href="cat_5.html#K2" title="CAT_5:func.2">`12</a> ,<font color="Maroon" title="c16">g9</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a></span>;<br/><a NAME="E27:8_1_5_3"/><span class="lab"><font color="Green" title="E40">A63</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c17">h9</font> <a href="cat_5.html#K1" title="CAT_5:func.1">`11</a> ,<font color="Maroon" title="c17">h9</font> <a href="cat_5.html#K2" title="CAT_5:func.2">`12</a> ,<font color="Maroon" title="c17">h9</font> <a href="cat_5.html#K3" title="CAT_5:func.3">`2</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a></span>;<br/><span class="kw">thus </span>the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:8_1"><span class="lab"><font color="Green" title="E14">A26</font></span></a>, <a class="txt" href="#E14:8_1_5_3"><span class="lab"><font color="Green" title="E28">A51</font></span></a>, <a class="txt" href="#E18:8_1_5_3"><span class="lab"><font color="Green" title="E32">A55</font></span></a>, <a class="txt" href="#E22:8_1_5_3"><span class="lab"><font color="Green" title="E36">A59</font></span></a></span>
<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">h9</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">g9</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">f9</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:8_1_5_3"><span class="lab"><font color="Green" title="E26">A49</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<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">h9</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">g9</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">f9</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E1:8_1_5_3"><span class="lab"><font color="Green" title="E17">A40</font></span></a>, <a class="txt" href="#E2:8_1_5_3"><span class="lab"><font color="Green" title="E18">A41</font></span></a>, <a class="txt" href="#E3:8_1_5_3"><span class="lab"><font color="Green" title="E19">A42</font></span></a>, <a class="txt" href="#E4:8_1_5_3"><span class="lab"><font color="Green" title="E20">A43</font></span></a>, <a class="txt" href="#E5:8_1_5_3"><span class="lab"><font color="Green" title="E21">A44</font></span></a>, <a class="txt" href="#E6:8_1_5_3"><span class="lab"><font color="Green" title="E22">A45</font></span></a>, <a class="txt" href="#E25:8_1_5_3"><span class="lab"><font color="Green" title="E38">A61</font></span></a>, <a class="txt" href="#E26:8_1_5_3"><span class="lab"><font color="Green" title="E39">A62</font></span></a>, <a class="txt" href="#E27:8_1_5_3"><span class="lab"><font color="Green" title="E40">A63</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8_1_5_3"><span class="lab"><font color="Green" title="E27">A50</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:8_1"><span class="lab"><font color="Green" title="E14">A26</font></span></a>, <a class="txt" href="#E17:8_1_5_3"><span class="lab"><font color="Green" title="E31">A54</font></span></a>, <a class="txt" href="#E21:8_1_5_3"><span class="lab"><font color="Green" title="E35">A58</font></span></a>, <a class="txt" href="#E23:8_1_5_3"><span class="lab"><font color="Green" title="E37">A60</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><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> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> &amp; the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</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> #)  holds <br/>(  not the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</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> #)  holds <br/>(  not the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> ) ) )</span><br/>

<span class="kw">consider </span><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>()<span class="kw"> such that </span><br/><a NAME="E5:8_1_5"/><span class="lab"><font color="Green" title="E17">A64</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:8_1_5"/>
<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>]
 <span class="kw">and </span><br/><a NAME="E7:8_1_5"/><span class="lab"><font color="Green" title="E18">A65</font></span>: 
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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:8_1"><span class="lab"><font color="Green" title="E16">A30</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">b9</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/>
<span class="kw">reconsider </span><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">b9</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">M</font> ;<br/>
<span class="kw">thus </span>the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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#U2" title="CAT_1:sel.2">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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_5"><span class="lab"><font color="Green" title="E17">A64</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</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> #)  holds <br/>(  not the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</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> #)  holds <br/>(  not the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> ) ) )</span><br/>

<span class="kw">thus </span>the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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#U2" title="CAT_1:sel.2">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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:8_1"><span class="lab"><font color="Green" title="E11">A11</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_5"><span class="lab"><font color="Green" title="E17">A64</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</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> #)  holds <br/>(  not the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</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> #)  holds <br/>(  not the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> ) ) )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</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> #)  holds <br/>(  not the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c14">f9</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> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> implies the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U2" title="CAT_1:sel.2">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">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f9</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c15">g</font> = <font color="Maroon" title="c14">f9</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">M</font> ;<br/><span class="kw">assume </span><a NAME="E2:8_1_5_6"/>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</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">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U2" title="CAT_1:sel.2">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">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f9</font></span><br/><a NAME="E3:8_1_5_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A66</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:8_1"><span class="lab"><font color="Green" title="E11">A11</font></span></a></span>;<br/><a NAME="E4:8_1_5_6"/><span class="lab"><font color="Green" title="E20">A67</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_5"><span class="lab"><font color="Green" title="E17">A64</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/><a NAME="E5:8_1_5_6"/><span class="lab"><font color="Green" title="E21">A68</font></span>: 
<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> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E3:8_1_5_6"><span class="lab"><font color="Green" title="E19">A66</font></span></a></span>;<br/><a NAME="E6:8_1_5_6"/><span class="lab"><font color="Green" title="E22">A69</font></span>: 
<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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">CC</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8_1"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E3:8_1_5_6"><span class="lab"><font color="Green" title="E19">A66</font></span></a>, <a class="txt" href="#E4:8_1_5_6"><span class="lab"><font color="Green" title="E20">A67</font></span></a></span>;<br/><a NAME="E7:8_1_5_6"/><span class="lab"><font color="Green" title="E23">A70</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_5"><span class="lab"><font color="Green" title="E17">A64</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/><a NAME="E8:8_1_5_6"/><span class="lab"><font color="Green" title="E24">A71</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_5"><span class="lab"><font color="Green" title="E17">A64</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/><a NAME="E9:8_1_5_6"/><span class="lab"><font color="Green" title="E25">A72</font></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>) <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:8_1_5"><span class="lab"><font color="Green" title="E18">A65</font></span></a>, <a class="txt" href="#E5:8_1_5_6"><span class="lab"><font color="Green" title="E21">A68</font></span></a></span>;<br/><a NAME="E10:8_1_5_6"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:8_1"><span class="lab"><font color="Green" title="E14">A26</font></span></a>, <a class="txt" href="#E6:8_1_5_6"><span class="lab"><font color="Green" title="E22">A69</font></span></a>, <a class="txt" href="#E7:8_1_5_6"><span class="lab"><font color="Green" title="E23">A70</font></span></a>, <a class="txt" href="#E8:8_1_5_6"><span class="lab"><font color="Green" title="E24">A71</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:8_1_5_6"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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#U2" title="CAT_1:sel.2">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">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E3:8_1_5_6"><span class="lab"><font color="Green" title="E19">A66</font></span></a>, <a class="txt" href="#E9:8_1_5_6"><span class="lab"><font color="Green" title="E25">A72</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c14">f9</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> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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> <font color="Maroon" title="c14">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> or the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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">f9</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</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">f9</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c15">g</font> = <font color="Maroon" title="c14">f9</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">M</font> ;<br/>
<span class="kw">assume </span><a NAME="E14:8_1_5"/>
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</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">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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">f9</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">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_1.html#K1" title="FUNCT_1:func.1">.</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">f9</font></span><br/>

<a NAME="E15:8_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A73</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a></span>;<br/>
<a NAME="E16:8_1_5"/><span class="lab"><font color="Green" title="E20">A74</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_5"><span class="lab"><font color="Green" title="E17">A64</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/>
<a NAME="E17:8_1_5"/><span class="lab"><font color="Green" title="E21">A75</font></span>: 
<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> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E15:8_1_5"><span class="lab"><font color="Green" title="E19">A73</font></span></a></span>;<br/>
<a NAME="E18:8_1_5"/><span class="lab"><font color="Green" title="E22">A76</font></span>: 
<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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">CC</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8_1"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E15:8_1_5"><span class="lab"><font color="Green" title="E19">A73</font></span></a>, <a class="txt" href="#E16:8_1_5"><span class="lab"><font color="Green" title="E20">A74</font></span></a></span>;<br/>
<a NAME="E19:8_1_5"/><span class="lab"><font color="Green" title="E23">A77</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_5"><span class="lab"><font color="Green" title="E17">A64</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/>
<a NAME="E20:8_1_5"/><span class="lab"><font color="Green" title="E24">A78</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_5"><span class="lab"><font color="Green" title="E17">A64</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E21:8_1_5"/><span class="lab"><font color="Green" title="E25">A79</font></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>) <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:8_1_5"><span class="lab"><font color="Green" title="E18">A65</font></span></a>, <a class="txt" href="#E17:8_1_5"><span class="lab"><font color="Green" title="E21">A75</font></span></a></span>;<br/>
<a NAME="E22:8_1_5"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:8_1"><span class="lab"><font color="Green" title="E14">A26</font></span></a>, <a class="txt" href="#E18:8_1_5"><span class="lab"><font color="Green" title="E22">A76</font></span></a>, <a class="txt" href="#E19:8_1_5"><span class="lab"><font color="Green" title="E23">A77</font></span></a>, <a class="txt" href="#E20:8_1_5"><span class="lab"><font color="Green" title="E24">A78</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:8_1_5"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon">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">f9</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">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">f9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E15:8_1_5"><span class="lab"><font color="Green" title="E19">A73</font></span></a>, <a class="txt" href="#E21:8_1_5"><span class="lab"><font color="Green" title="E25">A79</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="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="E22:8_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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="cat_5.html#D1" target="_self" title="CAT_5:def.1">CAT_5:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <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="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c11">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">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="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><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/>
<span class="kw">then </span><span class="kw">consider </span><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>()<span class="kw"> such that </span><br/><a NAME="E3:8_1_6"/><span class="lab"><font color="Green" title="E17">A80</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:8_1_6"/>
<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/>
<span class="kw">take </span>
<font color="Maroon" title="c14">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c11">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">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></span><br/>

<span class="lab"><font color="Green" title="E18">A81</font></span>:  <a href="graph_1.html#K1" title="GRAPH_1:func.1">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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_6"><span class="lab"><font color="Green" title="E17">A80</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>
;<br/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:8_1"><span class="lab"><font color="Green" title="E11">A11</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_6"><span class="lab"><font color="Green" title="E17">A80</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:8_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="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c11">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_6"><span class="lab"><font color="Green" title="E17">A80</font></span></a>, <a class="txt" href="#E5:8_1_6"><span class="lab"><font color="Green" title="E18">A81</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="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/>
<span class="kw">take </span>
<font color="Maroon" title="c11">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <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>()<br/> for <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 <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>] holds <br/><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> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">C</font> ) &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">C</font>  ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">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="b4">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="Olive" title="b1">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="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b4">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="b2">a</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">f</font>] ) ) &amp; ( for <font color="Olive" title="b1">m1</font>, <font color="Olive" title="b2">m2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">C</font><br/> for <font color="Olive" title="b3">a1</font>, <font color="Olive" title="b4">a2</font>, <font color="Olive" title="b5">a3</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="b6">f1</font>, <font color="Olive" title="b7">f2</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 <font color="Olive" title="b1">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="Olive" title="b3">a1</font>,<font color="Olive" title="b4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b6">f1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b2">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="Olive" title="b4">a2</font>,<font color="Olive" title="b5">a3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b7">f2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> holds <br/><font color="Olive" title="b2">m2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">m1</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="Olive" title="b3">a1</font>,<font color="Olive" title="b5">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="Olive" title="b7">f2</font>,<font color="Olive" title="b6">f1</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E24:8_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="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">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 <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>] holds <br/><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> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">C</font> ) &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">C</font>  ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">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="b4">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="Olive" title="b1">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="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b4">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="b2">a</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">f</font>] ) ) &amp; ( for <font color="Olive" title="b1">m1</font>, <font color="Olive" title="b2">m2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">C</font><br/> for <font color="Olive" title="b3">a1</font>, <font color="Olive" title="b4">a2</font>, <font color="Olive" title="b5">a3</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="b6">f1</font>, <font color="Olive" title="b7">f2</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 <font color="Olive" title="b1">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="Olive" title="b3">a1</font>,<font color="Olive" title="b4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b6">f1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b2">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="Olive" title="b4">a2</font>,<font color="Olive" title="b5">a3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b7">f2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> holds <br/><font color="Olive" title="b2">m2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">m1</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="Olive" title="b3">a1</font>,<font color="Olive" title="b5">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="Olive" title="b7">f2</font>,<font color="Olive" title="b6">f1</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">m</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">C</font>  ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">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="b4">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="Olive" title="b1">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="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b4">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="b2">a</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">f</font>] ) ) &amp; ( for <font color="Olive" title="b1">m1</font>, <font color="Olive" title="b2">m2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">C</font><br/> for <font color="Olive" title="b3">a1</font>, <font color="Olive" title="b4">a2</font>, <font color="Olive" title="b5">a3</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="b6">f1</font>, <font color="Olive" title="b7">f2</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 <font color="Olive" title="b1">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="Olive" title="b3">a1</font>,<font color="Olive" title="b4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b6">f1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b2">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="Olive" title="b4">a2</font>,<font color="Olive" title="b5">a3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b7">f2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> holds <br/><font color="Olive" title="b2">m2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">m1</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="Olive" title="b3">a1</font>,<font color="Olive" title="b5">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="Olive" title="b7">f2</font>,<font color="Olive" title="b6">f1</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span>
<div class="add"><span class="kw">let </span><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>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font>,<font color="Olive" title="b1">f</font>] holds <br/><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="Olive" title="b1">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></span><br/><span class="kw">let </span><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>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>] implies <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> )</span><br/><span class="kw">assume </span><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>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><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></span><br/><a NAME="E2:8_1_7"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">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/><span class="kw">hence </span><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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m1</font>, <font color="Olive" title="b2">m2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">C</font><br/> for <font color="Olive" title="b3">a1</font>, <font color="Olive" title="b4">a2</font>, <font color="Olive" title="b5">a3</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="b6">f1</font>, <font color="Olive" title="b7">f2</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 <font color="Olive" title="b1">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="Olive" title="b3">a1</font>,<font color="Olive" title="b4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b6">f1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b2">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="Olive" title="b4">a2</font>,<font color="Olive" title="b5">a3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b7">f2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> holds <br/><font color="Olive" title="b2">m2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">m1</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="Olive" title="b3">a1</font>,<font color="Olive" title="b5">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="Olive" title="b7">f2</font>,<font color="Olive" title="b6">f1</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>
<div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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>] )</span><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/><span class="kw">hence </span><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>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font>, <font color="Olive" title="b3">a3</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="b4">f1</font>, <font color="Olive" title="b5">f2</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 <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="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b4">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="Olive" title="b2">a2</font>,<font color="Olive" title="b3">a3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b5">f2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> holds <br/><font color="Maroon" title="c13">m2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c12">m1</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="Olive" title="b1">a1</font>,<font color="Olive" title="b3">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="Olive" title="b5">f2</font>,<font color="Olive" title="b4">f1</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/><span class="kw">let </span><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>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</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 <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="Olive" title="b1">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="Olive" title="b2">f2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> holds <br/><font color="Maroon" title="c13">m2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c12">m1</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="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="Olive" title="b2">f2</font>,<font color="Olive" title="b1">f1</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/><span class="kw">let </span><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>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> implies <font color="Maroon" title="c13">m2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c12">m1</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="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> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E27:8_1"/><span class="lab"><font color="Green" title="E17">A82</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E28:8_1"/><span class="lab"><font color="Green" title="E18">A83</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">m2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c12">m1</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="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></span><br/>

<a NAME="E29:8_1"/><span class="lab"><font color="Green" title="E19">A84</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:8_1"><span class="lab"><font color="Green" title="E17">A82</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/>
<a NAME="E30:8_1"/><span class="lab"><font color="Green" title="E20">A85</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:8_1"><span class="lab"><font color="Green" title="E17">A82</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/>
<a NAME="E31:8_1"/><span class="lab"><font color="Green" title="E21">A86</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:8_1"><span class="lab"><font color="Green" title="E18">A83</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/>
<a NAME="E32:8_1"/><span class="lab"><font color="Green" title="E22">A87</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:8_1"><span class="lab"><font color="Green" title="E18">A83</font></span></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></span>;<br/>
<a NAME="E33:8_1"/><span class="lab"><font color="Green" title="E23">A88</font></span>: 
<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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">CC</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8_1"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E30:8_1"><span class="lab"><font color="Green" title="E20">A85</font></span></a>, <a class="txt" href="#E31:8_1"><span class="lab"><font color="Green" title="E21">A86</font></span></a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c13">m2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D4" title="CAT_1:def.4">CAT_1:def 4</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:8_1"><span class="lab"><font color="Green" title="E14">A26</font></span></a>, <a class="txt" href="#E29:8_1"><span class="lab"><font color="Green" title="E19">A84</font></span></a>, <a class="txt" href="#E32:8_1"><span class="lab"><font color="Green" title="E22">A87</font></span></a>, <a class="txt" href="#E33:8_1"><span class="lab"><font color="Green" title="E23">A88</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:8_1"><span class="lab"><font color="Green" title="E18">A83</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:8_1"><span class="lab"><font color="Green" title="E17">A82</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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