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

<b>reconsider </b><font color="Maroon">f</font> =  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="closure2.html#K6" target="_self">Bool</a> <font color="Maroon">M</font></span>)</span> as   <a href="closure2.html#NM3">SetOp</a> of <font color="Maroon">M</font> ;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E2:53_1_1"/>
<font color="Maroon">f</font> is <a href="closure2.html#V7" target="_self">reflexive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6" target="_self">Bool</a> <font color="Maroon">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D12" target="_self">CLOSURE2:def 12</a><br/>

<b>thus </b><a NAME="E1:53_1_1_1"/>
<font color="Maroon">X</font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">f</font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>


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

<b>thus </b><a NAME="E3:53_1_1"/>
<font color="Maroon">f</font> is <a href="closure2.html#V8" target="_self">monotonic</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6" target="_self">Bool</a> <font color="Maroon">M</font>, <font color="Maroon">Y</font> be    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6" target="_self">Bool</a> <font color="Maroon">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D13" target="_self">CLOSURE2:def 13</a><br/>

<b>assume </b><a NAME="E1:53_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:53_1_1_2"/>
( <font color="Maroon">f</font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">X</font> <a href="pboole.html#R6">=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f</font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">Y</font> <a href="pboole.html#R6">=</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E3:53_1_1_2"/>
<font color="Maroon">f</font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">X</font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">f</font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E4:53_1_1"/>
<font color="Maroon">f</font> is <a href="closure2.html#V9" target="_self">idempotent</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6" target="_self">Bool</a> <font color="Maroon">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D14" target="_self">CLOSURE2:def 14</a><br/>

<b>thus </b><a NAME="E1:53_1_1_3"/>
<font color="Maroon">f</font> <a href="closure2.html#K7" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">X</font></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">f</font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>


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

<b>thus </b><a NAME="E5:53_1_1"/>
<font color="Maroon">f</font> is <a href="closure2.html#V10" target="_self">topological</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6" target="_self">Bool</a> <font color="Maroon">M</font>, <font color="Maroon">Y</font> be    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6" target="_self">Bool</a> <font color="Maroon">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D15" target="_self">CLOSURE2:def 15</a><br/>

<a NAME="E1:53_1_1_4"/>
<font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">Y</font> is    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:53_1_1_4_1"/>
( <font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">M</font> &amp; <font color="Maroon">Y</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<b>hence </b><a NAME="E2:53_1_1_4_1"/>
<font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">Y</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="pboole.html#T18">PBOOLE:18</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a><br/>


</div><b>end;</b></div>
<a NAME="E2:53_1_1_4"/><b>then </b>
<font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="closure2.html#K6" target="_self">Bool</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span> = 
<font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">Y</font>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">X</font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">Y</font>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">X</font></span>)</span> <a href="pboole.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/><br/>


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


</div>
