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

<b>set </b><font color="Maroon" title="c2">c</font> =  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font>,   <a href="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> $1,$2;<br/>
<b>consider </b><font color="Maroon" title="c3">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="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">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="E2:1_1_1"/><i><font color="Green" title="E1">A1</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> holds  <font color="Maroon" title="c3">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="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">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">f1</font>, <font color="Olive" title="b5">f2</font>, <font color="Olive" title="b6">g1</font>, <font color="Olive" title="b7">g2</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">g1</font>,<font color="Olive" title="b7">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">g1</font>,<font color="Olive" title="b7">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></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> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b6">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b4">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b5">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) );<br/>
<a NAME="E3:1_1_1"/><i><font color="Green" title="E2">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="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">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="c3">M</font>,<font color="Maroon" title="c3">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="c3">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="1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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="c4">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="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">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="c3">M</font>,<font color="Maroon" title="c3">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="c4">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="c3">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="c4">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="c4">ijk</font>] ) )</span><br/>

<b>assume </b><a NAME="E1:1_1_1_1"/>
<font color="Maroon" title="c4">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="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">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="c3">M</font>,<font color="Maroon" title="c3">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="c4">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="c3">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="c4">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="c4">ijk</font>] )</span><br/>

<b>then consider </b><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font>, <font color="Maroon" title="c7">x3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:1_1_1_1"/><i><font color="Green" title="E3">A3</font></i>: 
( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c7">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c4">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="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">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="c8">x1</font> = <font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c9">x2</font> = <font color="Maroon" title="c6">x2</font>, <font color="Maroon" title="c10">x3</font> = <font color="Maroon" title="c7">x3</font> as    <a href="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="msinst_1.html#E3:1_1_1_1"><i><font color="Green" title="E3">A3</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c11">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="c11">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="c3">M</font>,<font color="Maroon" title="c3">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="c4">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="c3">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="c4">ijk</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c4">ijk</font>] ) )</span><br/>

<b>assume </b><a NAME="E5:1_1_1_1"/><i><font color="Green" title="E4">A4</font></i>: 
<font color="Maroon" title="c11">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="c3">M</font>,<font color="Maroon" title="c3">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="c4">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="c3">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="c4">ijk</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c4">ijk</font>] )</span><br/>

<a NAME="E6:1_1_1_1"/><i><font color="Green" title="E5">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="c3">M</font>,<font color="Maroon" title="c3">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">x1</font>,<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">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="c3">M</font>,<font color="Maroon" title="c3">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">x1</font>,<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">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="E7:1_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="c3">M</font>,<font color="Maroon" title="c3">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">x1</font>,<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">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="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">x3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">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="E8:1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c11">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="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">x3</font> &amp; <font color="Maroon" title="c11">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="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font> )
 
<b>by </b><i><a class="txt" href="msinst_1.html#E3:1_1_1_1"><i><font color="Green" title="E3">A3</font></i></a>, <a class="txt" href="msinst_1.html#E5:1_1_1_1"><i><font color="Green" title="E4">A4</font></i></a>, <a class="txt" href="msinst_1.html#E6:1_1_1_1"><i><font color="Green" title="E5">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="E9:1_1_1_1"/><b>then </b><i><font color="Green" title="E6">A6</font></i>: 
( <font color="Maroon" title="c11">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="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">x3</font> &amp; <font color="Maroon" title="c11">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="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font> )
 
<b>by </b><i><a class="txt" href="msinst_1.html#E2:1_1_1"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">f1</font>, <font color="Maroon" title="c13">f2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E11:1_1_1_1"/><i><font color="Green" title="E7">A7</font></i>: 
( <font color="Maroon" title="c11">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> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">f1</font>,<font color="Maroon" title="c13">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">f1</font>,<font color="Maroon" title="c13">f2</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font> )
 <b>by </b><i><a class="ref" href="msalimit.html#D10" title="MSALIMIT:def.10">MSALIMIT:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">g1</font>, <font color="Maroon" title="c15">g2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E13:1_1_1_1"/><i><font color="Green" title="E8">A8</font></i>: 
( <font color="Maroon" title="c11">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> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">g1</font>,<font color="Maroon" title="c15">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">g1</font>,<font color="Maroon" title="c15">g2</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">x3</font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E9:1_1_1_1"><i><font color="Green" title="E6">A6</font></i></a>, <a class="ref" href="msalimit.html#D10" title="MSALIMIT:def.10">MSALIMIT:def 10</a></i>;<br/>
<b>take </b><font color="Maroon" title="c16">y</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">f2</font></span>)</span></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="c16">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="c3">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="c4">ijk</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c4">ijk</font>] )</span><br/>

<a NAME="E14:1_1_1_1"/><i><font color="Green" title="E9">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="c3">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="c4">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="c3">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="c8">x1</font>,<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">x3</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E3:1_1_1_1"><i><font color="Green" title="E3">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="E15:1_1_1_1"/><i><font color="Green" title="E10">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="c3">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="c8">x1</font>,<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c10">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/>
<a NAME="E16:1_1_1_1"/>
<font color="Maroon" title="c14">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">f1</font>,<font color="Maroon" title="c15">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">f2</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c10">x3</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E11:1_1_1_1"><i><font color="Green" title="E7">A7</font></i></a>, <a class="txt" href="msinst_1.html#E13:1_1_1_1"><i><font color="Green" title="E8">A8</font></i></a>, <a class="ref" href="pua2mss1.html#T30" title="PUA2MSS1:th.30">PUA2MSS1:30</a></i>;<br/>
<a NAME="E17:1_1_1_1"/><b>then </b>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c10">x3</font>
 
<b>by </b><i><a class="ref" href="msalimit.html#D10" title="MSALIMIT:def.10">MSALIMIT:def 10</a></i>;<br/>
<b>hence </b><a NAME="E18:1_1_1_1"/>
<font color="Maroon" title="c16">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="c3">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="c4">ijk</font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E2:1_1_1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="msinst_1.html#E14:1_1_1_1"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="msinst_1.html#E15:1_1_1_1"><i><font color="Green" title="E10">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="c16">y</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c4">ijk</font>]</span><br/>

<b>take </b>
<font color="Maroon" title="c8">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="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c4">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="c8">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">f1</font>, <font color="Olive" title="b4">f2</font>, <font color="Olive" title="b5">g1</font>, <font color="Olive" title="b6">g2</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">f1</font>,<font color="Olive" title="b4">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">x1</font>,<font color="Olive" title="b1">j</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">g1</font>,<font color="Olive" title="b6">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="c11">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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">g1</font>,<font color="Olive" title="b6">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">f1</font>,<font color="Olive" title="b4">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon" title="c16">y</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">(<span class="default"><font color="Olive" title="b5">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b3">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b6">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b4">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>

<b>take </b>
<font color="Maroon" title="c9">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="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c4">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="c8">x1</font>,<font color="Maroon" title="c9">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">f1</font>, <font color="Olive" title="b3">f2</font>, <font color="Olive" title="b4">g1</font>, <font color="Olive" title="b5">g2</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">f1</font>,<font color="Olive" title="b3">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">g1</font>,<font color="Olive" title="b5">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x2</font>,<font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c11">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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">g1</font>,<font color="Olive" title="b5">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">f1</font>,<font color="Olive" title="b3">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon" title="c16">y</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">(<span class="default"><font color="Olive" title="b4">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b3">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>

<b>take </b>
<font color="Maroon" title="c10">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="c4">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="c8">x1</font>,<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; ( for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font>, <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">g1</font>,<font color="Olive" title="b4">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">x3</font> &amp; <font color="Maroon" title="c11">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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">g1</font>,<font color="Olive" title="b4">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon" title="c16">y</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">(<span class="default"><font color="Olive" title="b3">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>

<b>thus </b><a NAME="E19:1_1_1_1"/>
<font color="Maroon" title="c4">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="c8">x1</font>,<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <b>by </b><i><a class="txt" href="msinst_1.html#E3:1_1_1_1"><i><font color="Green" title="E3">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">f1</font>, <font color="Olive" title="b2">f2</font>, <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">g1</font>,<font color="Olive" title="b4">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">x3</font> &amp; <font color="Maroon" title="c11">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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">g1</font>,<font color="Olive" title="b4">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon" title="c16">y</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">(<span class="default"><font color="Olive" title="b3">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<b>let </b><font color="Maroon" title="c17">F1</font>, <font color="Maroon" title="c18">F2</font>, <font color="Maroon" title="c19">G1</font>, <font color="Maroon" title="c20">G2</font> be   <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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F1</font>,<font color="Maroon" title="c18">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">G1</font>,<font color="Maroon" title="c20">G2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">x3</font> &amp; <font color="Maroon" title="c11">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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">G1</font>,<font color="Maroon" title="c20">G2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F1</font>,<font color="Maroon" title="c18">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> implies <font color="Maroon" title="c16">y</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">(<span class="default"><font color="Maroon" title="c19">G1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c17">F1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">G2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c18">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<b>assume </b><a NAME="E20:1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F1</font>,<font color="Maroon" title="c18">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">G1</font>,<font color="Maroon" title="c20">G2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">x3</font> &amp; <font color="Maroon" title="c11">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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">G1</font>,<font color="Maroon" title="c20">G2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F1</font>,<font color="Maroon" title="c18">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></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="c16">y</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">(<span class="default"><font color="Maroon" title="c19">G1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c17">F1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">G2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c18">F2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<a NAME="E21:1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c11">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> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">G1</font>,<font color="Maroon" title="c20">G2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">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> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F1</font>,<font color="Maroon" title="c18">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E22:1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c19">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g1</font> &amp; <font color="Maroon" title="c20">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g2</font> &amp; <font color="Maroon" title="c17">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> &amp; <font color="Maroon" title="c18">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f2</font> )
 
<b>by </b><i><a class="txt" href="msinst_1.html#E11:1_1_1_1"><i><font color="Green" title="E7">A7</font></i></a>, <a class="txt" href="msinst_1.html#E13:1_1_1_1"><i><font color="Green" title="E8">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E23:1_1_1_1"/>
<font color="Maroon" title="c16">y</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">(<span class="default"><font color="Maroon" title="c19">G1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c17">F1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">G2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c18">F2</font></span>)</span></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"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">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="c3">M</font>,<font color="Maroon" title="c3">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="c3">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span><b> such that </b><br/><a NAME="E5:1_1_1"/><i><font color="Green" title="E3">A11</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="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">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="c3">M</font>,<font color="Maroon" title="c3">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="c3">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="c4">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="c3">M</font>,<font color="Maroon" title="c3">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="msinst_1.html#E3:1_1_1"><i><font color="Green" title="E2">A2</font></i></a>);<br/></i>
<b>take </b><font color="Maroon" title="c5">EX</font> =  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">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="c5">EX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> holds  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c5">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="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <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="c5">EX</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> holds <br/>for <font color="Olive" title="b4">f1</font>, <font color="Olive" title="b5">f2</font>, <font color="Olive" title="b6">g1</font>, <font color="Olive" title="b7">g2</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c5">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; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">g1</font>,<font color="Olive" title="b7">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c5">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="c5">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> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">g1</font>,<font color="Olive" title="b7">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</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"><span class="p2">(<span class="default"><font color="Olive" title="b6">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b4">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b5">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c6">EX</font> = <font color="Maroon" title="c5">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="E7:1_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="c6">EX</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> holds <br/>for <font color="Olive" title="b4">f1</font>, <font color="Olive" title="b5">f2</font>, <font color="Olive" title="b6">g1</font>, <font color="Olive" title="b7">g2</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">g1</font>,<font color="Olive" title="b7">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">g1</font>,<font color="Olive" title="b7">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</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"><span class="p2">(<span class="default"><font color="Olive" title="b6">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b4">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b5">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c8">j</font>, <font color="Maroon" title="c9">k</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c6">EX</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> implies for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font>, <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">g1</font>,<font color="Olive" title="b4">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">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="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">g1</font>,<font color="Olive" title="b4">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</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"><span class="p2">(<span class="default"><font color="Olive" title="b3">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:1_1_1_2"/>
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</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">f1</font>, <font color="Olive" title="b2">f2</font>, <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">g1</font>,<font color="Olive" title="b4">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">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="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">g1</font>,<font color="Olive" title="b4">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</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"><span class="p2">(<span class="default"><font color="Olive" title="b3">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<b>let </b><font color="Maroon" title="c10">f1</font>, <font color="Maroon" title="c11">f2</font>, <font color="Maroon" title="c12">g1</font>, <font color="Maroon" title="c13">g2</font> be   <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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">f1</font>,<font color="Maroon" title="c11">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">g1</font>,<font color="Maroon" title="c13">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">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="c6">EX</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_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">g1</font>,<font color="Maroon" title="c13">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">f1</font>,<font color="Maroon" title="c11">f2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c12">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<b>assume </b><a NAME="E2:1_1_1_2"/><i><font color="Green" title="E4">A12</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">f1</font>,<font color="Maroon" title="c11">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">g1</font>,<font color="Maroon" title="c13">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">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="c6">EX</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_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">g1</font>,<font color="Maroon" title="c13">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">f1</font>,<font color="Maroon" title="c11">f2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c12">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<b>set </b><font color="Maroon" title="c14">ijk</font> = <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>;<br/>
<a NAME="E3:1_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="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">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="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">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="c15">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="c3">M</font>,<font color="Maroon" title="c3">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="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">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="c3">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="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span><b> such that </b><br/><a NAME="E5:1_1_1_2"/><i><font color="Green" title="E5">A13</font></i>: 
( <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">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="c3">M</font>,<font color="Maroon" title="c3">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="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">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="c15">f</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="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>] ) )
 <b>by </b><i><a class="txt" href="msinst_1.html#E5:1_1_1"><i><font color="Green" title="E3">A11</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">x</font> = <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"><font color="Maroon" title="c12">g1</font>,<font color="Maroon" title="c13">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">f1</font>,<font color="Maroon" title="c11">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E6:1_1_1_2"/><i><font color="Green" title="E6">A14</font></i>: 
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c3">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="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">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="c3">M</font>,<font color="Maroon" title="c3">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="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">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="E7:1_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="c3">M</font>,<font color="Maroon" title="c3">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="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">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="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">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="E8:1_1_1_2"/><b>then </b>
<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"><font color="Maroon" title="c12">g1</font>,<font color="Maroon" title="c13">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">f1</font>,<font color="Maroon" title="c11">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></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="c3">M</font>,<font color="Maroon" title="c3">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="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E2:1_1_1_2"><i><font color="Green" title="E4">A12</font></i></a>, <a class="txt" href="msinst_1.html#E6:1_1_1_2"><i><font color="Green" title="E6">A14</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="c17">I</font>, <font color="Maroon" title="c18">J</font>, <font color="Maroon" title="c19">K</font> being    <a href="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E10:1_1_1_2"/><i><font color="Green" title="E7">A15</font></i>: 
( <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">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="c17">I</font>,<font color="Maroon" title="c18">J</font>,<font color="Maroon" title="c19">K</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; ( for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font>, <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c17">I</font>,<font color="Maroon" title="c18">J</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">g1</font>,<font color="Olive" title="b4">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c18">J</font>,<font color="Maroon" title="c19">K</font> &amp; <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"><font color="Maroon" title="c12">g1</font>,<font color="Maroon" title="c13">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">f1</font>,<font color="Maroon" title="c11">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">g1</font>,<font color="Olive" title="b4">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon" title="c15">f</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">g1</font>,<font color="Maroon" title="c13">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">f1</font>,<font color="Maroon" title="c11">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></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"><span class="p2">(<span class="default"><font color="Olive" title="b3">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
 <b>by </b><i><a class="txt" href="msinst_1.html#E5:1_1_1_2"><i><font color="Green" title="E5">A13</font></i></a></i>;<br/>
<a NAME="E11:1_1_1_2"/><i><font color="Green" title="E8">A16</font></i>: 
( <font color="Maroon" title="c17">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c18">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c19">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> )
 
<b>by </b><i><a class="txt" href="msinst_1.html#E10:1_1_1_2"><i><font color="Green" title="E7">A15</font></i></a>, <a class="ref" href="mcart_1.html#T28" title="MCART_1:th.28">MCART_1:28</a></i>;<br/>
<a NAME="E12:1_1_1_2"/>
<font color="Maroon" title="c15">f</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="c6">EX</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E5:1_1_1_2"><i><font color="Green" title="E5">A13</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="E13:1_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="c6">EX</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_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">g1</font>,<font color="Maroon" title="c13">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">f1</font>,<font color="Maroon" title="c11">f2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c12">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="msinst_1.html#E2:1_1_1_2"><i><font color="Green" title="E4">A12</font></i></a>, <a class="txt" href="msinst_1.html#E10:1_1_1_2"><i><font color="Green" title="E7">A15</font></i></a>, <a class="txt" href="msinst_1.html#E11:1_1_1_2"><i><font color="Green" title="E8">A16</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="E8:1_1_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">EX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> holds  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c5">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="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <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="c5">EX</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> holds <br/>for <font color="Olive" title="b4">f1</font>, <font color="Olive" title="b5">f2</font>, <font color="Olive" title="b6">g1</font>, <font color="Olive" title="b7">g2</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c5">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; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">g1</font>,<font color="Olive" title="b7">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c5">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="c5">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> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">g1</font>,<font color="Olive" title="b7">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</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"><span class="p2">(<span class="default"><font color="Olive" title="b6">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b4">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b5">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
 <b>by </b><i><a class="txt" href="msinst_1.html#E2:1_1_1"><i><font color="Green" title="E1">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>
