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

<b>reconsider </b><font color="Maroon">F</font> =  <a href="msualg_3.html#K2">id</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font></span>)</span> as   <a href="closure1.html#NM1">MSSetOp</a> of <font color="Maroon">M</font> ;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E2:20_1_1"/>
<font color="Maroon">F</font> is <a href="closure1.html#V1" target="_self">reflexive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure1.html#D2" target="_self">CLOSURE1:def 2</a><br/>

<b>thus </b><a NAME="E1:20_1_1_1"/>
<font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">F</font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E3:20_1_1"/>
<font color="Maroon">F</font> is <a href="closure1.html#V2" target="_self">monotonic</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font>, <font color="Maroon">Y</font> be    <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure1.html#D3" target="_self">CLOSURE1:def 3</a><br/>

<b>assume </b><a NAME="E1:20_1_1_2"/>
<font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">Y</font>
 ;<br/>

<b>hence </b><a NAME="E2:20_1_1_2"/>
<font color="Maroon">F</font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">F</font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E4:20_1_1"/>
<font color="Maroon">F</font> is <a href="closure1.html#V3" target="_self">idempotent</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure1.html#D4" target="_self">CLOSURE1:def 4</a><br/>

<b>thus </b><a NAME="E1:20_1_1_3"/>
<font color="Maroon">F</font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">X</font> <a href="pboole.html#R6">=</a> <font color="Maroon">F</font> <a href="closure1.html#K3" target="_self">..</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E5:20_1_1"/>
<font color="Maroon">F</font> is <a href="closure1.html#V4" target="_self">topological</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font>, <font color="Maroon">Y</font> be    <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure1.html#D5" target="_self">CLOSURE1:def 5</a><br/>

<a NAME="E1:20_1_1_4"/>
( <font color="Maroon">X</font> <a href="pboole.html#R1">in</a>  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font> &amp; <font color="Maroon">Y</font> <a href="pboole.html#R1">in</a>  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="ref" href="mssubfam.html#T12">MSSUBFAM:12</a></i>;<br/>
<a NAME="E2:20_1_1_4"/><b>then </b>
( <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="mboolean.html#T1">MBOOLEAN:1</a></i>;<br/>
<a NAME="E3:20_1_1_4"/><b>then </b>
<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>;<br/>
<a NAME="E4:20_1_1_4"/><b>then </b>
<font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">Y</font> <a href="pboole.html#R1">in</a>  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="mboolean.html#T1">MBOOLEAN:1</a></i>;<br/>
<a NAME="E5:20_1_1_4"/><b>then </b>
<font color="Maroon">X</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">Y</font> is    <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T11">MSSUBFAM:11</a></i>;<br/>
<b>hence </b><a NAME="E6:20_1_1_4"/>
<font color="Maroon">F</font> <a href="closure1.html#K1" target="_self">..</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> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="closure1.html#K3" 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="closure1.html#K3" target="_self">..</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i/>;<br/>


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


</div>
