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

<b>set </b><font color="Maroon">c</font> =  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="msalimit.html#M3">Element</a> of  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font>,   <a href="msalimit.html#M3">Element</a> of  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="msalimit.html#K6">MSS_morph</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">M</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E2:1_1_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="msalimit.html#M3">Element</a> of  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> holds  <font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>,<font color="Olive">j</font>)
 <b>from </b><i><a class="ref" href="altcat_1.html#S2">ALTCAT_1:sch 2</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">i</font>, <font color="Olive">j</font>, <font color="Olive">k</font> being   <a href="msalimit.html#M3">Element</a> of  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> st <br/>( <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">k</font></span><a href="mcart_1.html#K3">]</a></span> &amp; ( for <font color="Olive">f1</font>, <font color="Olive">f2</font>, <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="funct_1.html#NM1">Function</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">f1</font>,<font color="Olive">f2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g1</font>,<font color="Olive">g2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Olive">j</font>,<font color="Olive">k</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g1</font>,<font color="Olive">g2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">f1</font>,<font color="Olive">f2</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">g1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">f2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) );<br/>
<a NAME="E3:1_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">ijk</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">ijk</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span> holds <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">ijk</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">M</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">ijk</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>,<font color="Olive">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">ijk</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_1_1"/>
<font color="Maroon">ijk</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> &amp; <font color="Maroon">ijk</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="mcart_1.html#K3">]</a></span> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T72">MCART_1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font>, <font color="Maroon">x2</font> = <font color="Maroon">x2</font>, <font color="Maroon">x3</font> = <font color="Maroon">x3</font> as    <a href="msalimit.html#M3">Element</a> of  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E5:1_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ijk</font>
 ;<br/>

<a NAME="E6:1_1_1_1"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="mcart_1.html#K3">]</a></span>
 
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<a NAME="E7:1_1_1_1"/>
<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">x3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D6">ALTCAT_1:def 6</a></i>;<br/>
<a NAME="E8:1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">x3</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> )
 
<b>by </b><i>, , , <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E9:1_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K6">MSS_morph</a> <font color="Maroon">x2</font>,<font color="Maroon">x3</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K6">MSS_morph</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">f2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:1_1_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">f1</font>,<font color="Maroon">f2</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> )
 <b>by </b><i><a class="ref" href="msalimit.html#D10">MSALIMIT:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">g1</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E13:1_1_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">g2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">g1</font>,<font color="Maroon">g2</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">x2</font>,<font color="Maroon">x3</font> )
 <b>by </b><i>, <a class="ref" href="msalimit.html#D10">MSALIMIT:def 10</a></i>;<br/>
<b>take </b><font color="Maroon">y</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/>

<a NAME="E14:1_1_1_1"/><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">M</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ijk</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">M</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>
 
<b>by </b><i>, <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<a NAME="E15:1_1_1_1"/><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">M</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x3</font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D5">ALTCAT_1:def 5</a></i>;<br/>
<a NAME="E16:1_1_1_1"/>
<font color="Maroon">g1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font>,<font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f2</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">x1</font>,<font color="Maroon">x3</font>
 
<b>by </b><i>, , <a class="ref" href="pua2mss1.html#T30">PUA2MSS1:30</a></i>;<br/>
<a NAME="E17:1_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K6">MSS_morph</a> <font color="Maroon">x1</font>,<font color="Maroon">x3</font>
 
<b>by </b><i><a class="ref" href="msalimit.html#D10">MSALIMIT:def 10</a></i>;<br/>
<b>hence </b><a NAME="E18:1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">M</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ijk</font>
 <b>by </b><i>, , </i>;<br/>

<b>take </b>
<font color="Maroon">x1</font>
;<br/>

<b>take </b>
<font color="Maroon">x2</font>
;<br/>

<b>take </b>
<font color="Maroon">x3</font>
;<br/>

<b>thus </b><a NAME="E19:1_1_1_1"/>
<font color="Maroon">ijk</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="mcart_1.html#K3">]</a></span>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">F1</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">F2</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">G1</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">G2</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E20:1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">x3</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> )
 ;<br/>

<a NAME="E21:1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E22:1_1_1_1"/><b>then </b>
( <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E23:1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="altcat_1.html#K3">|}</a></span>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">M</font></span><a href="altcat_1.html#K2">|}</a></span><b> such that </b><br/><a NAME="E5:1_1_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">ijk</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">ijk</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span> holds <br/> ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">ijk</font>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">M</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">ijk</font> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">ijk</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">ijk</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">x</font>,<font color="Olive">ijk</font>] ) )
 <b>from </b><i><a class="ref" href="mssubfam.html#S1">MSSUBFAM:sch 1</a>();<br/></i>
<b>take </b><font color="Maroon">EX</font> =  <a href="altcat_1.html#G2">AltCatStr</a>(# <span class="p1">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">M</font>,<font color="Maroon">F</font> #);<br/>

<b>reconsider </b><font color="Maroon">EX</font> = <font color="Maroon">EX</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="altcat_1.html#L2">AltCatStr</a>  ;<br/>
<a NAME="E7:1_1_1"/>
for <font color="Olive">i</font>, <font color="Olive">j</font>, <font color="Olive">k</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">EX</font>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> holds <br/>for <font color="Olive">f1</font>, <font color="Olive">f2</font>, <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="funct_1.html#NM1">Function</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">f1</font>,<font color="Olive">f2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">EX</font> <a href="binop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g1</font>,<font color="Olive">g2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">EX</font> <a href="binop_1.html#K1">.</a> <font color="Olive">j</font>,<font color="Olive">k</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">EX</font> <a href="altcat_1.html#K4">.</a> <font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">k</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g1</font>,<font color="Olive">g2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">f1</font>,<font color="Olive">f2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">g1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">f2</font></span>)</span></span><a href="tarski.html#K4">]</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">i</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">EX</font>, <font color="Maroon">j</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">EX</font>, <font color="Maroon">k</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">EX</font>;<br/>

<b>assume </b><a NAME="E1:1_1_1_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> )
 ;<br/>

<b>let </b><font color="Maroon">f1</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">f2</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g1</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g2</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E2:1_1_1_2"/><i><font color="Green">E58</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">EX</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">g2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">EX</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font> )
 ;<br/>

<b>set </b><font color="Maroon">ijk</font> = <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span>;<br/>
<a NAME="E3:1_1_1_2"/>
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T73">MCART_1:73</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">M</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span><b> such that </b><br/><a NAME="E5:1_1_1_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">x</font>,<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span>] ) )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">x</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">g2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E6:1_1_1_2"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span>
 
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<a NAME="E7:1_1_1_2"/>
<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D6">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">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">g2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">I</font> being    <a href="msalimit.html#M3">Element</a> of  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font>, <font color="Maroon">J</font> being    <a href="msalimit.html#M3">Element</a> of  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font>, <font color="Maroon">K</font> being    <a href="msalimit.html#M3">Element</a> of  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E10:1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">J</font>,<font color="Maroon">K</font></span><a href="mcart_1.html#K3">]</a></span> &amp; ( for <font color="Olive">f1</font>, <font color="Olive">f2</font>, <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="funct_1.html#NM1">Function</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">f1</font>,<font color="Olive">f2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">I</font>,<font color="Maroon">J</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g1</font>,<font color="Olive">g2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">J</font>,<font color="Maroon">K</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">g2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g1</font>,<font color="Olive">g2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">f1</font>,<font color="Olive">f2</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">g2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">g1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">f2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) )
 <b>by </b><i/>;<br/>
<a NAME="E11:1_1_1_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">J</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T28">MCART_1:28</a></i>;<br/>
<a NAME="E12:1_1_1_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">EX</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="multop_1.html#D1">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">Comp</a> of <font color="Maroon">EX</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">g2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:1_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">EX</font> <a href="hidden.html#R1">=</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="msalimit.html#M3">Element</a> of  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> holds  the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">EX</font> <a href="binop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="msalimit.html#K6">MSS_morph</a> <font color="Olive">i</font>,<font color="Olive">j</font> ) &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</font>, <font color="Olive">k</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">EX</font>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> holds <br/>for <font color="Olive">f1</font>, <font color="Olive">f2</font>, <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="funct_1.html#NM1">Function</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">f1</font>,<font color="Olive">f2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">EX</font> <a href="binop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g1</font>,<font color="Olive">g2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">EX</font> <a href="binop_1.html#K1">.</a> <font color="Olive">j</font>,<font color="Olive">k</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">EX</font> <a href="altcat_1.html#K4">.</a> <font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">k</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g1</font>,<font color="Olive">g2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">f1</font>,<font color="Olive">f2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">g1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">f2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) )
 <b>by </b><i/>;<br/>


</div>
