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

<b>reconsider </b><font color="Maroon" title="c3">f</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font></span>)</span> as   <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of <font color="Maroon" title="c2">M</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c3">f</font>
;<br/>

<b>thus </b><a NAME="E2:23_1_1"/>
<font color="Maroon" title="c3">f</font> is <a href="closure3.html#V1" title="CLOSURE3:attr.1">algebraic</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="closure3.html#M1" title="CLOSURE3:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure3.html#D5" target="_self" title="CLOSURE3:def.5">CLOSURE3:def 5</a><br/>

<b>assume </b><a NAME="E1:23_1_1_1"/>
<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="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">x</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is    <a href="closure3.html#M1" title="CLOSURE3:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font> : ( <font color="Olive" title="b1">a</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> &amp;  <a href="closure3.html#K1" title="CLOSURE3:func.1">supp</a> <font color="Olive" title="b1">a</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">a</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">x</font> ) </span>}</span> ;<br/>
<a NAME="E2:23_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is    <a href="closure3.html#M1" title="CLOSURE3:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font> : ( <font color="Olive" title="b1">a</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> &amp;  <a href="closure3.html#K1" title="CLOSURE3:func.1">supp</a> <font color="Olive" title="b1">a</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">a</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">x</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1_1_1_1"/>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is    <a href="closure3.html#M1" title="CLOSURE3:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font> : ( <font color="Olive" title="b1">a</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> &amp;  <a href="closure3.html#K1" title="CLOSURE3:func.1">supp</a> <font color="Olive" title="b1">a</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">a</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">x</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">a</font> being    <a href="closure3.html#M1" title="CLOSURE3:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E3:23_1_1_1_1"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c7">a</font>
 <b>and </b><br/><a NAME="E4:23_1_1_1_1"/>
( <font color="Maroon" title="c7">a</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> &amp;  <a href="closure3.html#K1" title="CLOSURE3:func.1">supp</a> <font color="Maroon" title="c7">a</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c7">a</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">x</font> )
 ;<br/>
<b>thus </b><a NAME="E5:23_1_1_1_1"/>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="closure3.html#E3:23_1_1_1_1"><i><font color="Green" title="E8">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is    <a href="closure3.html#M1" title="CLOSURE3:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font> : ( <font color="Olive" title="b1">a</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> &amp;  <a href="closure3.html#K1" title="CLOSURE3:func.1">supp</a> <font color="Olive" title="b1">a</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">a</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">x</font> ) </span>}</span>  as   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c6">A</font>
;<br/>

<b>thus </b><a NAME="E4:23_1_1_1"/>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is    <a href="closure3.html#M1" title="CLOSURE3:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font> : ( <font color="Olive" title="b1">a</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> &amp;  <a href="closure3.html#K1" title="CLOSURE3:func.1">supp</a> <font color="Olive" title="b1">a</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Olive" title="b1">a</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">x</font> ) </span>}</span> 
 ;<br/>

<b>thus </b><a NAME="E5:23_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c6">A</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D13" title="PBOOLE:def.13">PBOOLE:def 13</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:23_1_1_1_2"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:23_1_1_1_2"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">a</font> being    <a href="closure3.html#M1" title="CLOSURE3:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E4:23_1_1_1_2"/><i><font color="Green" title="E10">A4</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c9">a</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> &amp;  <a href="closure3.html#K1" title="CLOSURE3:func.1">supp</a> <font color="Maroon" title="c9">a</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c9">a</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="txt" href="closure3.html#E1:23_1_1_1_2"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="closure3.html#E2:23_1_1_1_2"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="closure3.html#T9" target="_self" title="CLOSURE3:th.9">Th9</a></i>;<br/>
<a NAME="E5:23_1_1_1_2"/>
<font color="Maroon" title="c9">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c9">a</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<a NAME="E6:23_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 
<b>by </b><i><a class="txt" href="closure3.html#E4:23_1_1_1_2"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/>
<a NAME="E7:23_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="closure3.html#M1" title="CLOSURE3:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> </span>}</span> 
 
;<br/>
<a NAME="E8:23_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="closure3.html#M1" title="CLOSURE3:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E4:23_1_1_1_2"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:23_1_1_1_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="closure3.html#E1:23_1_1_1_2"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="closure3.html#D4" target="_self" title="CLOSURE3:def.4">Def4</a></i>;<br/>


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

<b>thus </b><a NAME="E6:23_1_1_1"/>
 <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c6">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:23_1_1_1_3"/><i><font color="Green" title="E8">A5</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<a NAME="E2:23_1_1_1_3"/>
for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> holds <br/><font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23_1_1_1_3_1"/>
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 ;<br/>

<a NAME="E2:23_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> where <font color="Olive" title="b1">ff</font> is    <a href="closure3.html#M1" title="CLOSURE3:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">ff</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E1:23_1_1_1_3"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="closure3.html#D4" target="_self" title="CLOSURE3:def.4">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:23_1_1_1_3_1"/><i><font color="Green" title="E9">A6</font></i>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font>
 <b>and </b><br/><a NAME="E5:23_1_1_1_3_1"/><i><font color="Green" title="E10">A7</font></i>: 
<font color="Maroon" title="c9">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> where <font color="Olive" title="b1">ff</font> is    <a href="closure3.html#M1" title="CLOSURE3:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">ff</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">ff</font> being    <a href="closure3.html#M1" title="CLOSURE3:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E7:23_1_1_1_3_1"/><i><font color="Green" title="E11">A8</font></i>: 
<font color="Maroon" title="c9">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>and </b><br/><a NAME="E8:23_1_1_1_3_1"/><i><font color="Green" title="E12">A9</font></i>: 
<font color="Maroon" title="c10">ff</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 <b>by </b><i><a class="txt" href="closure3.html#E5:23_1_1_1_3_1"><i><font color="Green" title="E10">A7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">a</font> being    <a href="closure3.html#M1" title="CLOSURE3:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E10:23_1_1_1_3_1"/><i><font color="Green" title="E13">A10</font></i>: 
<font color="Maroon" title="c10">ff</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c11">a</font>
 <b>and </b><br/><a NAME="E11:23_1_1_1_3_1"/><i><font color="Green" title="E14">A11</font></i>: 
( <font color="Maroon" title="c11">a</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> &amp;  <a href="closure3.html#K1" title="CLOSURE3:func.1">supp</a> <font color="Maroon" title="c11">a</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c11">a</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="txt" href="closure3.html#E8:23_1_1_1_3_1"><i><font color="Green" title="E12">A9</font></i></a></i>;<br/>
<a NAME="E12:23_1_1_1_3_1"/>
<font color="Maroon" title="c10">ff</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c11">a</font>
 
<b>by </b><i><a class="txt" href="closure3.html#E10:23_1_1_1_3_1"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<a NAME="E13:23_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c10">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 
<b>by </b><i><a class="txt" href="closure3.html#E1:23_1_1_1_3"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="closure3.html#E11:23_1_1_1_3_1"><i><font color="Green" title="E14">A11</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<b>hence </b><a NAME="E14:23_1_1_1_3_1"/>
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="closure3.html#E4:23_1_1_1_3_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="closure3.html#E7:23_1_1_1_3_1"><i><font color="Green" title="E11">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:23_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


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


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

<b>thus </b><a NAME="E3:23_1_1"/>
<font color="Maroon" title="c3">f</font> is <a href="closure2.html#V7" title="CLOSURE2:attr.7">reflexive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">X</font> be    <a href="closure3.html#M1" title="CLOSURE3:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D12" title="CLOSURE2:def.12">CLOSURE2:def 12</a><br/>

<b>thus </b><a NAME="E1:23_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="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>


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

<b>thus </b><a NAME="E4:23_1_1"/>
<font color="Maroon" title="c3">f</font> is <a href="closure2.html#V8" title="CLOSURE2:attr.8">monotonic</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c5">Y</font> be    <a href="closure3.html#M1" title="CLOSURE3:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D13" title="CLOSURE2:def.13">CLOSURE2:def 13</a><br/>

<b>assume </b><a NAME="E1:23_1_1_3"/><i><font color="Green" title="E8">A12</font></i>: 
<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>
 ;<br/>

<a NAME="E2:23_1_1_3"/>
( <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">Y</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E3:23_1_1_3"/>
<font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</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="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="closure3.html#E1:23_1_1_3"><i><font color="Green" title="E8">A12</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E5:23_1_1"/>
<font color="Maroon" title="c3">f</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">X</font> be    <a href="closure3.html#M1" title="CLOSURE3:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D14" title="CLOSURE2:def.14">CLOSURE2:def 14</a><br/>

<b>thus </b><a NAME="E1:23_1_1_4"/>
<font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>


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


</div>
