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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><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">o</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/>







<b>assume </b><a NAME="E1:76"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> holds  <font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Olive">b</font>,<font color="Olive">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D13">BINOP_1:def 13</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:76_1"/>
<font color="Maroon">o</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">o</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:76_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:76_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">o</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:76_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="txt" href="monoid_1.html#E4:76_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E6:76_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="domain_1.html#K3">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E7:76_1_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="domain_1.html#K3">`2</a> </span>)</span> &amp; <font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="domain_1.html#K3">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="domain_1.html#K2">`1</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="monoid_1.html#E3:76_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="monoid_1.html#E4:76_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E8:76_1_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="domain_1.html#K3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E9:76_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:76"/>
( <font color="Maroon">o</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">o</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">o</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">o</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>


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