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

<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">I</font>, <font color="Maroon" title="c3">J</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c5">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c6">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c7">X</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">I</font>;<br/><b>let </b><font color="Maroon" title="c8">Y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c3">J</font>;<br/>













<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E22">A1</font></i>: 
( ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font> holds  <font color="Maroon" title="c6">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c7">X</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3" title="CAT_2:func.3">curry'</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> ) &amp; <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  &amp; <font color="Maroon" title="c4">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; <font color="Maroon" title="c4">F</font> <a href="finseqop.html#NR2" title="FINSEQOP:NR.2">has_an_inverseOp</a>  )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c3">J</font>] <b>means </b><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c7">X</font>,$1</span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> $1,<font color="Maroon" title="c6">g</font>;<br/>
<a NAME="E2:42"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c3">J</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c9">T</font> =  <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ;<br/>
<a NAME="E2:42_1"/>
<font color="Maroon" title="c9">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c7">X</font>,<span class="p2">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></i>;<br/>
<a NAME="E3:42_1"/><b>then </b>
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c7">X</font>,<span class="p2">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c4">F</font>
 
<b>by </b><i><a class="txt" href="matrix_3.html#E1:42"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="setwiseo.html#T40" title="SETWISEO:th.40">SETWISEO:40</a></i>;<br/>
<b>hence </b><a NAME="E4:42_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c3">J</font>]
 <b>by </b><i><a class="txt" href="matrix_3.html#E1:42"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="setwiseo.html#T40" title="SETWISEO:th.40">SETWISEO:40</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:42"/><i><font color="Green" title="E24">A3</font></i>: 
for <font color="Olive" title="b1">Y1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c3">J</font><br/> for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Y1</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Y1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">Y1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c3">J</font>;<br/><b>let </b><font color="Maroon" title="c10">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">J</font>;<br/>



<b>assume </b><a NAME="E1:42_2"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c9">Y1</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">Y1</font>,<font color="Maroon" title="c6">g</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">s</font> = <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c3">J</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_2_1"/>
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y1</font> or  not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1_1"><b>suppose </b></a><a NAME="E1:42_2_1_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y1</font>
 ;<br/><div class="add"><a NAME="E2:42_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">Y1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Y1</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>;<br/><b>hence </b><a NAME="E3:42_2_1_1"/>
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c7">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">Y1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Y1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c6">g</font>
 <b>by </b><i><a class="txt" href="matrix_3.html#E1:42_2"><i><font color="Green" title="E25">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1_2"><b>suppose </b></a><a NAME="E1:42_2_1_2"/>
not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y1</font>
 ;<br/><div class="add"><a NAME="E2:42_2_1_2"/><b>then </b><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c9">Y1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></i>;<br/><a NAME="E3:42_2_1_2"/><b>then </b><i><font color="Green" title="E27">A6</font></i>: 
<span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c9">Y1</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c11">s</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T127" title="ZFMISC_1:th.127">ZFMISC_1:127</a></i>;<br/><b>thus </b><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c7">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">Y1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<font color="Maroon" title="c5">f</font> = 
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c9">Y1</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c11">s</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span></span>)</span>,<font color="Maroon" title="c5">f</font>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T120" title="ZFMISC_1:th.120">ZFMISC_1:120</a></i>
<br/>.= 
<font color="Maroon" title="c4">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p2"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c9">Y1</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<font color="Maroon" title="c5">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p2"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c11">s</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<font color="Maroon" title="c5">f</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E1:42"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="matrix_3.html#E3:42_2_1_2"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="setwop_2.html#T6" title="SETWOP_2:th.6">SETWOP_2:6</a></i>
<br/>.= 
<font color="Maroon" title="c4">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">Y1</font>,<font color="Maroon" title="c6">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c6">g</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E1:42"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="matrix_3.html#E1:42_2"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="matrix_3.html#T33" target="_self" title="MATRIX_3:th.33">Th33</a></i>
<br/>.= 
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Y1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c6">g</font>
<b>by </b><i><a class="txt" href="matrix_3.html#E1:42"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="matrix_3.html#E2:42_2_1_2"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="setwop_2.html#T6" title="SETWOP_2:th.6">SETWOP_2:6</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:42"/>
for <font color="Olive" title="b1">Y1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c3">J</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Y1</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S4" title="SETWISEO:sch.4">SETWISEO:sch 4</a>(<a class="txt" href="matrix_3.html#E2:42"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="matrix_3.html#E3:42"><i><font color="Green" title="E24">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:42"/>
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c8">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c6">g</font>
 ;<br/>


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