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<div class="add">

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">J</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">J</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">I</font>,<font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font>;<br/>













<b>assume </b><a NAME="E1:40"/><i><font color="Green">E43</font></i>: 
( ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">f</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Olive">i</font></span>)</span> ) &amp; <font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  &amp; <font color="Maroon">F</font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">F</font> is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">F</font> <a href="finseqop.html#NR2" target="_self">has_an_inverseOp</a>  )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">I</font>] <b>means </b><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">g</font>;<br/>
<a NAME="E2:40"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1">{}.</a> <font color="Maroon">I</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">T</font> =  <a href="setwiseo.html#K1">{}.</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">J</font></span><a href="zfmisc_1.html#K2">:]</a></span> as    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">J</font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<a NAME="E2:40_1"/>
<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <font color="Maroon">I</font></span>)</span>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/>
<a NAME="E3:40_1"/><b>then </b>
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <font color="Maroon">I</font></span>)</span>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="setwiseo.html#T40">SETWISEO:40</a></i>;<br/>
<b>hence </b><a NAME="E4:40_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1">{}.</a> <font color="Maroon">I</font>]
 <b>by </b><i>, <a class="ref" href="setwiseo.html#T40">SETWISEO:40</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:40"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">X1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">I</font><br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">X1</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">X1</font> <a href="finsub_1.html#K1">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Olive">x</font></span><a href="setwiseo.html#K2">.}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E1:40_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">X1</font>,<font color="Maroon">g</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">s</font> = <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span> as    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">I</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_1"><b>suppose </b></a><a NAME="E1:40_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<br/><div class="add"><a NAME="E2:40_2_1_1"/><b>then </b>
<font color="Maroon">X1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>;<br/><b>hence </b><a NAME="E3:40_2_1_1"/>
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="finsub_1.html#K1">\/</a> <span class="p3"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="finsub_1.html#K1">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<font color="Maroon">g</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_2"><b>suppose </b></a><a NAME="E1:40_2_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<br/><div class="add"><a NAME="E2:40_2_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">X1</font> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/><a NAME="E3:40_2_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T127">ZFMISC_1:127</a></i>;<br/><b>thus </b><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="finsub_1.html#K1">\/</a> <span class="p3"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<font color="Maroon">f</font> = 
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span> <a href="finsub_1.html#K1">\/</a> <span class="p2"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span></span>)</span>,<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T120">ZFMISC_1:120</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p2"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p2"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<font color="Maroon">f</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="setwop_2.html#T6">SETWOP_2:6</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">X1</font>,<font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">s</font>,<font color="Maroon">g</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="finsub_1.html#K1">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<font color="Maroon">g</font>
<b>by </b><i>, , <a class="ref" href="setwop_2.html#T6">SETWOP_2:6</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:40"/>
for <font color="Olive">X1</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">I</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">X1</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S4">SETWISEO:sch 4</a>(, );<br/></i>
<b>hence </b><a NAME="E5:40"/>
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">X</font>,<font color="Maroon">g</font>
 ;<br/>


</div>
