<?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">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="E2:24_1_1"/>
<font color="Maroon">f</font> is <a href="closure3.html#V1" target="_self">algebraic</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:24_1_1_1"/>
<font color="Maroon">x</font> <a href="pboole.html#R6">=</a> <font color="Maroon">f</font> <a href="closure2.html#K7">.</a> <font color="Maroon">x</font>
 ;<br/>

<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="closure2.html#K7">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font> : ( <font color="Olive">a</font> is <a href="finset_1.html#V2">locally-finite</a> &amp;  <a href="closure3.html#K1" target="_self">supp</a> <font color="Olive">a</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">a</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">x</font> ) </span>}</span> ;<br/>
<a NAME="E2:24_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="closure2.html#K7">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font> : ( <font color="Olive">a</font> is <a href="finset_1.html#V2">locally-finite</a> &amp;  <a href="closure3.html#K1" target="_self">supp</a> <font color="Olive">a</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">a</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">x</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:24_1_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="closure2.html#K7">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font> : ( <font color="Olive">a</font> is <a href="finset_1.html#V2">locally-finite</a> &amp;  <a href="closure3.html#K1" target="_self">supp</a> <font color="Olive">a</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">a</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">x</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font><b> such that </b><br/><a NAME="E3:24_1_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="closure2.html#K7">.</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:24_1_1_1_1"/>
( <font color="Maroon">a</font> is <a href="finset_1.html#V2">locally-finite</a> &amp;  <a href="closure3.html#K1" target="_self">supp</a> <font color="Maroon">a</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">a</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">x</font> )
 ;<br/>
<b>thus </b><a NAME="E5:24_1_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="closure2.html#K7">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font> : ( <font color="Olive">a</font> is <a href="finset_1.html#V2">locally-finite</a> &amp;  <a href="closure3.html#K1" target="_self">supp</a> <font color="Olive">a</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">a</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">x</font> ) </span>}</span>  as   <a href="closure2.html#NM1">SubsetFamily</a> of <font color="Maroon">M</font> ;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>thus </b><a NAME="E4:24_1_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="closure2.html#K7">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font> : ( <font color="Olive">a</font> is <a href="finset_1.html#V2">locally-finite</a> &amp;  <a href="closure3.html#K1" target="_self">supp</a> <font color="Olive">a</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">a</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">x</font> ) </span>}</span> 
 ;<br/>

<b>thus </b><a NAME="E5:24_1_1_1"/>
<font color="Maroon">x</font> <a href="pboole.html#R2">c=</a>  <a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">A</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D13">PBOOLE:def 13</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:24_1_1_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

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

<b>assume </b><a NAME="E2:24_1_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<b>consider </b><font color="Maroon">a</font> being    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font><b> such that </b><br/><a NAME="E4:24_1_1_1_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">a</font> is <a href="finset_1.html#V2">locally-finite</a> &amp;  <a href="closure3.html#K1" target="_self">supp</a> <font color="Maroon">a</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">a</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E5:24_1_1_1_2"/>
<font color="Maroon">a</font> <a href="pboole.html#R6">=</a> <font color="Maroon">f</font> <a href="closure2.html#K7">.</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<a NAME="E6:24_1_1_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:24_1_1_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">g</font> is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font> : <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 
;<br/>
<a NAME="E8:24_1_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">g</font> is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font> : <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:24_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E6:24_1_1_1"/>
 <a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">A</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:24_1_1_1_3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:24_1_1_1_3"/>
for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> holds <br/><font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:24_1_1_1_3_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E2:24_1_1_1_3_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">ff</font> is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font> : <font color="Olive">ff</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:24_1_1_1_3_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E5:24_1_1_1_3_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">ff</font> is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font> : <font color="Olive">ff</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">ff</font> being    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font><b> such that </b><br/><a NAME="E7:24_1_1_1_3_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E8:24_1_1_1_3_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">ff</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font><b> such that </b><br/><a NAME="E10:24_1_1_1_3_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">ff</font> <a href="pboole.html#R6">=</a> <font color="Maroon">f</font> <a href="closure2.html#K7">.</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E11:24_1_1_1_3_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">a</font> is <a href="finset_1.html#V2">locally-finite</a> &amp;  <a href="closure3.html#K1" target="_self">supp</a> <font color="Maroon">a</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">a</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">x</font> )
 <b>by </b><i/>;<br/>
<a NAME="E12:24_1_1_1_3_1"/>
<font color="Maroon">ff</font> <a href="pboole.html#R6">=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<a NAME="E13:24_1_1_1_3_1"/><b>then </b>
<font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<b>hence </b><a NAME="E14:24_1_1_1_3_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:24_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


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


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

<b>thus </b><a NAME="E3:24_1_1"/>
<font color="Maroon">f</font> is <a href="closure2.html#V7">reflexive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D12">CLOSURE2:def 12</a><br/>

<b>thus </b><a NAME="E1:24_1_1_2"/>
<font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">f</font> <a href="closure2.html#K7">.</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="E4:24_1_1"/>
<font color="Maroon">f</font> is <a href="closure2.html#V8">monotonic</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font>, <font color="Maroon">Y</font> be    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D13">CLOSURE2:def 13</a><br/>

<b>assume </b><a NAME="E1:24_1_1_3"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:24_1_1_3"/>
( <font color="Maroon">f</font> <a href="closure2.html#K7">.</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">.</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="E3:24_1_1_3"/>
<font color="Maroon">f</font> <a href="closure2.html#K7">.</a> <font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">f</font> <a href="closure2.html#K7">.</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E5:24_1_1"/>
<font color="Maroon">f</font> is <a href="closure2.html#V9">idempotent</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D14">CLOSURE2:def 14</a><br/>

<b>thus </b><a NAME="E1:24_1_1_4"/>
<font color="Maroon">f</font> <a href="closure2.html#K7">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="closure2.html#K7">.</a> <font color="Maroon">X</font></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">f</font> <a href="closure2.html#K7">.</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>


</div>
