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

<b>set </b><font color="Maroon" title="c3">c</font> =  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,$1,$2;<br/>
<b>consider </b><font color="Maroon" title="c4">M</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E2:12_1_1"/><i><font color="Green" title="E8">A1</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)
 <b>from </b><i><a class="ref" href="altcat_1.html#S2" title="ALTCAT_1:sch.2">ALTCAT_1:sch 2</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> st <br/>( $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; ( for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font> &amp; $2 <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"><font color="Olive" title="b5">g</font>,<font color="Olive" title="b4">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Olive" title="b4">f</font> ) );<br/>
<a NAME="E3:12_1_1"/><i><font color="Green" title="E9">A2</font></i>: 
for <font color="Olive" title="b1">ijk</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">ijk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> holds <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ijk</font> holds <br/> ex <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ijk</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">y</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">ijk</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">ijk</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">ijk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> implies for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">ijk</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">ijk</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">ijk</font>] ) )</span><br/>

<b>assume </b><a NAME="E1:12_1_1_1"/>
<font color="Maroon" title="c5">ijk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">ijk</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">ijk</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">ijk</font>] )</span><br/>

<b>then consider </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font>, <font color="Maroon" title="c8">x3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:12_1_1_1"/><i><font color="Green" title="E10">A3</font></i>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c5">ijk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T72" title="MCART_1:th.72">MCART_1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">x1</font> = <font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c10">x2</font> = <font color="Maroon" title="c7">x2</font>, <font color="Maroon" title="c11">x3</font> = <font color="Maroon" title="c8">x3</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="#E3:12_1_1_1"><i><font color="Green" title="E10">A3</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">ijk</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">ijk</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c5">ijk</font>] ) )</span><br/>

<b>assume </b><a NAME="E5:12_1_1_1"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">ijk</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">ijk</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c5">ijk</font>] )</span><br/>

<a NAME="E6:12_1_1_1"/><i><font color="Green" title="E12">A5</font></i>: 
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 
<b>by </b><i><a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>;<br/>
<a NAME="E7:12_1_1_1"/>
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">x3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D6" title="ALTCAT_1:def.6">ALTCAT_1:def 6</a></i>;<br/>
<a NAME="E8:12_1_1_1"/><b>then </b>
( <font color="Maroon" title="c12">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">x3</font> &amp; <font color="Maroon" title="c12">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font> )
 
<b>by </b><i><a class="txt" href="#E3:12_1_1_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="#E5:12_1_1_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="#E6:12_1_1_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<a NAME="E9:12_1_1_1"/><b>then </b><i><font color="Green" title="E13">A6</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">x3</font> &amp; <font color="Maroon" title="c12">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font> )
 
<b>by </b><i><a class="txt" href="#E2:12_1_1"><i><font color="Green" title="E8">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">M2</font>, <font color="Maroon" title="c14">N2</font> being  <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a> <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c15">g</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">M2</font>,<font color="Maroon" title="c14">N2</font><b> such that </b><br/><a NAME="E11:12_1_1_1"/><i><font color="Green" title="E14">A7</font></i>: 
( <font color="Maroon" title="c13">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font> &amp; <font color="Maroon" title="c14">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x3</font> &amp; <font color="Maroon" title="c15">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c13">M2</font> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c14">N2</font> &amp; <font color="Maroon" title="c15">g</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c13">M2</font>,<font color="Maroon" title="c14">N2</font> )
 <b>by </b><i><a class="ref" href="msinst_1.html#D3" target="_self" title="MSINST_1:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">M1</font>, <font color="Maroon" title="c17">N1</font> being  <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a> <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c18">f</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c16">M1</font>,<font color="Maroon" title="c17">N1</font><b> such that </b><br/><a NAME="E13:12_1_1_1"/><i><font color="Green" title="E15">A8</font></i>: 
( <font color="Maroon" title="c16">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x1</font> &amp; <font color="Maroon" title="c17">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2</font> &amp; <font color="Maroon" title="c18">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  &amp; the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c16">M1</font> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c17">N1</font> &amp; <font color="Maroon" title="c18">f</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c16">M1</font>,<font color="Maroon" title="c17">N1</font> )
 <b>by </b><i><a class="txt" href="#E9:12_1_1_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self" title="MSINST_1:def.3">Def3</a></i>;<br/>
<b>take </b><font color="Maroon" title="c19">y</font> = <font color="Maroon" title="c15">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c18">f</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">ijk</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c19">y</font>,<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c5">ijk</font>] )</span><br/>

<a NAME="E14:12_1_1_1"/><i><font color="Green" title="E16">A9</font></i>: 
<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">ijk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">x3</font>
 
<b>by </b><i><a class="txt" href="#E3:12_1_1_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>;<br/>
<a NAME="E15:12_1_1_1"/><i><font color="Green" title="E17">A10</font></i>: 
<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c11">x3</font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D5" title="ALTCAT_1:def.5">ALTCAT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">f</font> = <font color="Maroon" title="c18">f</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c16">M1</font>,<font color="Maroon" title="c13">M2</font> <b>by </b><i><a class="txt" href="#E11:12_1_1_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="#E13:12_1_1_1"><i><font color="Green" title="E15">A8</font></i></a></i>;<br/>
<a NAME="E17:12_1_1_1"/><i><font color="Green" title="E18">A11</font></i>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c16">M1</font> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c14">N2</font>
 
<b>by </b><i><a class="txt" href="#E11:12_1_1_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="#E13:12_1_1_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="autalg_1.html#T11" title="AUTALG_1:th.11">AUTALG_1:11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">fg</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c16">M1</font>,<font color="Maroon" title="c14">N2</font><b> such that </b><br/><a NAME="E19:12_1_1_1"/><i><font color="Green" title="E19">A12</font></i>: 
( <font color="Maroon" title="c21">fg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c20">f</font> &amp; <font color="Maroon" title="c21">fg</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c16">M1</font>,<font color="Maroon" title="c14">N2</font> )
 <b>by </b><i><a class="txt" href="#E11:12_1_1_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="#E13:12_1_1_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="msinst_1.html#T5" target="_self" title="MSINST_1:th.5">Th5</a></i>;<br/>
<a NAME="E20:12_1_1_1"/>
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c11">x3</font>
 
<b>by </b><i><a class="txt" href="#E11:12_1_1_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="#E13:12_1_1_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="txt" href="#E17:12_1_1_1"><i><font color="Green" title="E18">A11</font></i></a>, <a class="txt" href="#E19:12_1_1_1"><i><font color="Green" title="E19">A12</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self" title="MSINST_1:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E21:12_1_1_1"/>
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">ijk</font>
 <b>by </b><i><a class="txt" href="#E2:12_1_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="#E14:12_1_1_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="txt" href="#E15:12_1_1_1"><i><font color="Green" title="E17">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c19">y</font>,<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c5">ijk</font>]</span><br/>

<b>take </b>
<font color="Maroon" title="c9">x1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> st <br/>( <font color="Maroon" title="c5">ijk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Olive" title="b1">j</font>,<font color="Olive" title="b2">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; ( for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">g</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x1</font>,<font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">j</font>,<font color="Olive" title="b2">k</font> &amp; <font color="Maroon" title="c12">x</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"><font color="Olive" title="b4">g</font>,<font color="Olive" title="b3">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Olive" title="b3">f</font> ) )</span><br/>

<b>take </b>
<font color="Maroon" title="c10">x2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> st <br/>( <font color="Maroon" title="c5">ijk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font>,<font color="Olive" title="b1">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; ( for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font> &amp; <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">x2</font>,<font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c12">x</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"><font color="Olive" title="b3">g</font>,<font color="Olive" title="b2">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Olive" title="b2">f</font> ) )</span><br/>

<b>take </b>
<font color="Maroon" title="c11">x3</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">ijk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; ( for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">x3</font> &amp; <font color="Maroon" title="c12">x</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"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Olive" title="b1">f</font> ) )</span><br/>

<b>thus </b><a NAME="E22:12_1_1_1"/>
<font color="Maroon" title="c5">ijk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <b>by </b><i><a class="txt" href="#E3:12_1_1_1"><i><font color="Green" title="E10">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">x3</font> &amp; <font color="Maroon" title="c12">x</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"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Olive" title="b1">f</font></span><br/>

<b>let </b><font color="Maroon" title="c22">F</font>, <font color="Maroon" title="c23">G</font> be  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c22">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font> &amp; <font color="Maroon" title="c23">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">x3</font> &amp; <font color="Maroon" title="c12">x</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"><font color="Maroon" title="c23">G</font>,<font color="Maroon" title="c22">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> implies <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">G</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c22">F</font> )</span><br/>

<b>assume </b><a NAME="E23:12_1_1_1"/>
( <font color="Maroon" title="c22">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font> &amp; <font color="Maroon" title="c23">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">x3</font> &amp; <font color="Maroon" title="c12">x</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"><font color="Maroon" title="c23">G</font>,<font color="Maroon" title="c22">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">G</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c22">F</font></span><br/>

<a NAME="E24:12_1_1_1"/><b>then </b>
( <font color="Maroon" title="c12">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">G</font> &amp; <font color="Maroon" title="c12">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="c22">F</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E25:12_1_1_1"/>
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">G</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c22">F</font>
 <b>by </b><i><a class="txt" href="#E11:12_1_1_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="#E13:12_1_1_1"><i><font color="Green" title="E15">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">F</font> being    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span>,<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span><b> such that </b><br/><a NAME="E5:12_1_1"/><i><font color="Green" title="E10">A13</font></i>: 
for <font color="Olive" title="b1">ijk</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">ijk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> holds <br/> ex <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ijk</font>,<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ijk</font> st <br/>( <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ijk</font> &amp; ( for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ijk</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">ijk</font>] ) )
 <b>from </b><i><a class="ref" href="mssubfam.html#S1" title="MSSUBFAM:sch.1">MSSUBFAM:sch 1</a>(<a class="txt" href="#E3:12_1_1"><i><font color="Green" title="E9">A2</font></i></a>);<br/></i>
<b>take </b><font color="Maroon" title="c6">EX</font> =  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1">(<span class="default"><a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">F</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">EX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> holds  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c6">EX</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> ) &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">k</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c6">EX</font><br/> for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c6">EX</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c6">EX</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c6">EX</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b5">g</font>,<font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Olive" title="b4">f</font> ) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c7">EX</font> = <font color="Maroon" title="c6">EX</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="altcat_1.html#L2" title="ALTCAT_1:struct.2">AltCatStr</a>  ;<br/>
<a NAME="E7:12_1_1"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">k</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c7">EX</font><br/> for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c7">EX</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c7">EX</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c7">EX</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b5">g</font>,<font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Olive" title="b4">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">i</font>, <font color="Maroon" title="c9">j</font>, <font color="Maroon" title="c10">k</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c7">EX</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c7">EX</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c7">EX</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c7">EX</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Olive" title="b1">f</font></span><br/><b>let </b><font color="Maroon" title="c11">f</font>, <font color="Maroon" title="c12">g</font> be  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c7">EX</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c7">EX</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font> implies <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c7">EX</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">g</font>,<font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c11">f</font> )</span><br/>



<b>assume </b><a NAME="E1:12_1_1_2"/><i><font color="Green" title="E11">A14</font></i>: 
( <font color="Maroon" title="c11">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c7">EX</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c7">EX</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c7">EX</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">g</font>,<font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c11">f</font></span><br/>

<b>set </b><font color="Maroon" title="c13">ijk</font> = <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>;<br/>
<a NAME="E2:12_1_1_2"/>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T73" title="MCART_1:th.73">MCART_1:73</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">ff</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>,<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span><b> such that </b><br/><a NAME="E4:12_1_1_2"/><i><font color="Green" title="E12">A15</font></i>: 
( <font color="Maroon" title="c14">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font>,<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>] ) )
 <b>by </b><i><a class="txt" href="#E5:12_1_1"><i><font color="Green" title="E10">A13</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">x</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">g</font>,<font color="Maroon" title="c11">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E5:12_1_1_2"/><i><font color="Green" title="E13">A16</font></i>: 
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 
<b>by </b><i><a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>;<br/>
<a NAME="E6:12_1_1_2"/>
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D6" title="ALTCAT_1:def.6">ALTCAT_1:def 6</a></i>;<br/>
<a NAME="E7:12_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">g</font>,<font color="Maroon" title="c11">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c4">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 
<b>by </b><i><a class="txt" href="#E1:12_1_1_2"><i><font color="Green" title="E11">A14</font></i></a>, <a class="txt" href="#E5:12_1_1_2"><i><font color="Green" title="E13">A16</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">I</font>, <font color="Maroon" title="c17">J</font>, <font color="Maroon" title="c18">K</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E9:12_1_1_2"/><i><font color="Green" title="E14">A17</font></i>: 
( <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font>,<font color="Maroon" title="c18">K</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; ( for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c17">J</font>,<font color="Maroon" title="c18">K</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">g</font>,<font color="Maroon" title="c11">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon" title="c14">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">g</font>,<font color="Maroon" title="c11">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Olive" title="b1">f</font> ) )
 <b>by </b><i><a class="txt" href="#E4:12_1_1_2"><i><font color="Green" title="E12">A15</font></i></a></i>;<br/>
<a NAME="E10:12_1_1_2"/><i><font color="Green" title="E15">A18</font></i>: 
( <font color="Maroon" title="c16">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c17">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c18">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> )
 
<b>by </b><i><a class="txt" href="#E9:12_1_1_2"><i><font color="Green" title="E14">A17</font></i></a>, <a class="ref" href="mcart_1.html#T28" title="MCART_1:th.28">MCART_1:28</a></i>;<br/>
<a NAME="E11:12_1_1_2"/>
<font color="Maroon" title="c14">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c7">EX</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font>
 
<b>by </b><i><a class="txt" href="#E4:12_1_1_2"><i><font color="Green" title="E12">A15</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:12_1_1_2"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c7">EX</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">g</font>,<font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c11">f</font>
 <b>by </b><i><a class="txt" href="#E1:12_1_1_2"><i><font color="Green" title="E11">A14</font></i></a>, <a class="txt" href="#E9:12_1_1_2"><i><font color="Green" title="E14">A17</font></i></a>, <a class="txt" href="#E10:12_1_1_2"><i><font color="Green" title="E15">A18</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:12_1_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">EX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msinst_1.html#K2" title="MSINST_1:func.2">MSAlg_set</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font> holds  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c6">EX</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msinst_1.html#K3" title="MSINST_1:func.3">MSAlg_morph</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> ) &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">k</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c6">EX</font><br/> for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c6">EX</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c6">EX</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c6">EX</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b5">g</font>,<font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Olive" title="b4">f</font> ) )
 <b>by </b><i><a class="txt" href="#E2:12_1_1"><i><font color="Green" title="E8">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
