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

<span class="kw">reconsider </span><font color="Maroon" title="c3">F</font> =  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a> <font color="Maroon" title="c2">M</font></span>)</span> as   <a href="closure1.html#NM1" title="CLOSURE1:NM.1">MSSetOp</a> of <font color="Maroon" title="c2">M</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> is  <a href="closure1.html#V1" title="CLOSURE1:attr.1">reflexive</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="closure1.html#V2" title="CLOSURE1:attr.2">monotonic</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="closure1.html#V3" title="CLOSURE1:attr.3">idempotent</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="closure1.html#V4" title="CLOSURE1:attr.4">topological</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:20_1_1"/>
<font color="Maroon" title="c3">F</font> is  <a href="closure1.html#V1" title="CLOSURE1:attr.1">reflexive</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> is  <a href="closure1.html#V2" title="CLOSURE1:attr.2">monotonic</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="closure1.html#V3" title="CLOSURE1:attr.3">idempotent</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="closure1.html#V4" title="CLOSURE1:attr.4">topological</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">X</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">Element</a> of  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a> <font color="Maroon" title="c2">M</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="closure1.html#D2" target="_self" title="CLOSURE1:def.2">CLOSURE1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">F</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c4">X</font></span><br/>

<span class="kw">thus </span><a NAME="E1:20_1_1_1"/>
<font color="Maroon" title="c4">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">F</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="closure1.html#T8" target="_self" title="CLOSURE1:th.8">Th8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:20_1_1"/>
<font color="Maroon" title="c3">F</font> is  <a href="closure1.html#V2" title="CLOSURE1:attr.2">monotonic</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> is  <a href="closure1.html#V3" title="CLOSURE1:attr.3">idempotent</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="closure1.html#V4" title="CLOSURE1:attr.4">topological</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c5">Y</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">Element</a> of  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a> <font color="Maroon" title="c2">M</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="closure1.html#D3" target="_self" title="CLOSURE1:def.3">CLOSURE1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c5">Y</font> implies <font color="Maroon" title="c3">F</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c4">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">F</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c5">Y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_1_1_2"/>
<font color="Maroon" title="c4">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c5">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">F</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c4">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">F</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c5">Y</font></span><br/>

<span class="kw">hence </span><a NAME="E2:20_1_1_2"/>
<font color="Maroon" title="c3">F</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c4">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">F</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c5">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="closure1.html#T9" target="_self" title="CLOSURE1:th.9">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E4:20_1_1"/>
<font color="Maroon" title="c3">F</font> is  <a href="closure1.html#V3" title="CLOSURE1:attr.3">idempotent</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">F</font> is  <a href="closure1.html#V4" title="CLOSURE1:attr.4">topological</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">X</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">Element</a> of  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a> <font color="Maroon" title="c2">M</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="closure1.html#D4" target="_self" title="CLOSURE1:def.4">CLOSURE1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">F</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c4">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">F</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c4">X</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E1:20_1_1_3"/>
<font color="Maroon" title="c3">F</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c4">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">F</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c4">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="closure1.html#T8" target="_self" title="CLOSURE1:th.8">Th8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E5:20_1_1"/>
<font color="Maroon" title="c3">F</font> is  <a href="closure1.html#V4" title="CLOSURE1:attr.4">topological</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c5">Y</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">Element</a> of  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a> <font color="Maroon" title="c2">M</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="closure1.html#D5" target="_self" title="CLOSURE1:def.5">CLOSURE1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">Y</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c5">Y</font></span>)</span></span><br/>

<a NAME="E1:20_1_1_4"/>
<font color="Maroon" title="c5">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mssubfam.html#T12" title="MSSUBFAM:th.12">MSSUBFAM:12</a></span>;<br/>
<a NAME="E2:20_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c5">Y</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mboolean.html#T1" title="MBOOLEAN:th.1">MBOOLEAN:1</a></span>;<br/>
<a NAME="E3:20_1_1_4"/>
<font color="Maroon" title="c4">X</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mssubfam.html#T12" title="MSSUBFAM:th.12">MSSUBFAM:12</a></span>;<br/>
<a NAME="E4:20_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mboolean.html#T1" title="MBOOLEAN:th.1">MBOOLEAN:1</a></span>;<br/>
<a NAME="E5:20_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">Y</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_1_4"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="pboole.html#T18" title="PBOOLE:th.18">PBOOLE:18</a></span>;<br/>
<a NAME="E6:20_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mboolean.html#T1" title="MBOOLEAN:th.1">MBOOLEAN:1</a></span>;<br/>
<a NAME="E7:20_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">Y</font> is    <a href="pboole.html#M1" title="PBOOLE:mode.1">Element</a> of  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mssubfam.html#T11" title="MSSUBFAM:th.11">MSSUBFAM:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:20_1_1_4"/>
<font color="Maroon" title="c3">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">Y</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c5">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="closure1.html#T10" target="_self" title="CLOSURE1:th.10">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
