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

<b>let </b><font color="Maroon">D1</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">D2</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">f1</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D1</font>;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D2</font>;<br/>







<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> ] <b>means </b><span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="filter_1.html#K6" target="_self">:|</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="filter_1.html#K6" target="_self">:|</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>thus </b><a NAME="E1:37"/>
( <font color="Maroon">f1</font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">f2</font> is <a href="binop_1.html#V1">commutative</a> implies <span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="filter_1.html#K6" target="_self">:|</a></span> is <a href="binop_1.html#V1">commutative</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1"/><i><font color="Green">E68</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">D1</font> holds  <font color="Maroon">f1</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">f1</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/>

<b>assume </b><a NAME="E2:37_1"/><i><font color="Green">E69</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">D2</font> holds  <font color="Maroon">f2</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">f2</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/>

<a NAME="E3:37_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">d1</font>, <font color="Olive">d1'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D1</font><br/> for <font color="Olive">d2</font>, <font color="Olive">d2'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D2</font> holds  <font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">d1</font>,<font color="Olive">d2</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">d1'</font>,<font color="Olive">d2'</font></span><a href="domain_1.html#K1">]</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D1</font>, <font color="Maroon">b1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D1</font>;<br/><b>let </b><font color="Maroon">a2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D2</font>, <font color="Maroon">b2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D2</font>;<br/>



<b>thus </b><span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="filter_1.html#K6" target="_self">:|</a></span> <a href="binop_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="domain_1.html#K1">]</a></span> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b1</font>,<font color="Maroon">a1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b1</font>,<font color="Maroon">a1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b2</font>,<font color="Maroon">a2</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="filter_1.html#K6" target="_self">:|</a></span> <a href="binop_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:37_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <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">D1</font>,<font color="Maroon">D2</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>]
 <b>from </b><i><a class="ref" href="filter_1.html#S5" target="_self">FILTER_1:sch 5</a>();</i> <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D13">BINOP_1:def 13</a><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:37"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <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">D1</font>,<font color="Maroon">D2</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds  <span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="filter_1.html#K6" target="_self">:|</a></span> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="filter_1.html#K6" target="_self">:|</a></span> <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/>

<b>thus </b><a NAME="E3:37"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D1</font> holds  <font color="Maroon">f1</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">f1</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D1</font>;<br/><b>let </b><font color="Maroon">b1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D1</font>;<br/>



<b>consider </b><font color="Maroon">a2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D2</font>, <font color="Maroon">b2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D2</font>;<br/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> = 
<span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="filter_1.html#K6" target="_self">:|</a></span> <a href="binop_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="filter_1.html#K6" target="_self">:|</a></span> <a href="binop_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b1</font>,<font color="Maroon">a1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b2</font>,<font color="Maroon">a2</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:37_2"/>
<font color="Maroon">f1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b1</font>,<font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">a2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D13">BINOP_1:def 13</a><br/><b>let </b><font color="Maroon">b2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D2</font>;<br/>



<b>consider </b><font color="Maroon">a1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D1</font>, <font color="Maroon">b1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D1</font>;<br/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> = 
<span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="filter_1.html#K6" target="_self">:|</a></span> <a href="binop_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="filter_1.html#K6" target="_self">:|</a></span> <a href="binop_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b1</font>,<font color="Maroon">a1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b2</font>,<font color="Maroon">a2</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E6:37"/>
<font color="Maroon">f2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b2</font>,<font color="Maroon">a2</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div>
