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

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="closure2.html#K12">ClSys-&gt;ClOp</a> <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure3.html#D5" target="_self">CLOSURE3:def 5</a>,<a class="ref" href="closure3.html#D6" target="_self">CLOSURE3:def 6</a><br/>

<b>assume </b><a NAME="E1:29_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="closure2.html#K12">ClSys-&gt;ClOp</a> <span class="p2">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="closure2.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="closure2.html#K12">ClSys-&gt;ClOp</a> <span class="p4">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="closure2.html#K7">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> is <a href="pre_circ.html#V1">locally-finite</a> &amp;  <a href="closure3.html#K1" target="_self">supp</a> <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>5</sub></font> ) </span>}</span> ;<br/>
<a NAME="E2:29_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="closure2.html#K12">ClSys-&gt;ClOp</a> <span class="p4">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="closure2.html#K7">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> is <a href="pre_circ.html#V1">locally-finite</a> &amp;  <a href="closure3.html#K1" target="_self">supp</a> <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>5</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></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="E1:29_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="closure2.html#K12">ClSys-&gt;ClOp</a> <span class="p4">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="closure2.html#K7">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> is <a href="pre_circ.html#V1">locally-finite</a> &amp;  <a href="closure3.html#K1" target="_self">supp</a> <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>5</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:29_1_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="closure2.html#K12">ClSys-&gt;ClOp</a> <span class="p2">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="closure2.html#K7">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E4:29_1_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> is <a href="pre_circ.html#V1">locally-finite</a> &amp;  <a href="closure3.html#K1" target="_self">supp</a> <font color="Maroon">c<sub>8</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="closure3.html#E3:29_1_1_1"><i><font color="Green">E12</font></i></a></i>;<br/>
<a NAME="E6:29_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E7:29_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="closure2.html#K12">ClSys-&gt;ClOp</a> <span class="p4">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="closure2.html#K7">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> is <a href="pre_circ.html#V1">locally-finite</a> &amp;  <a href="closure3.html#K1" target="_self">supp</a> <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>5</sub></font> ) </span>}</span>  as   <a href="closure2.html#NM1">SubsetFamily</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>thus </b><a NAME="E5:29_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="closure2.html#K12">ClSys-&gt;ClOp</a> <span class="p4">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="closure2.html#K7">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : ( <font color="Olive">b<sub>1</sub></font> is <a href="pre_circ.html#V1">locally-finite</a> &amp;  <a href="closure3.html#K1" target="_self">supp</a> <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>5</sub></font> ) </span>}</span> 
 ;<br/>

<b>thus </b><a NAME="E6:29_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a>  <a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">c<sub>7</sub></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="29_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:29_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:29_1_1_2_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></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="E2:29_1_1_2_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:29_1_1_2_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E5:29_1_1_2_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> is <a href="pre_circ.html#V1">locally-finite</a> &amp;  <a href="closure3.html#K1" target="_self">supp</a> <font color="Maroon">c<sub>11</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="closure3.html#E1:29_1_1_2_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="closure3.html#E2:29_1_1_2_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="closure3.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E6:29_1_1_2_1"/><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><a href="closure2.html#K12">ClSys-&gt;ClOp</a> <span class="p2">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="closure2.html#K7">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="closure3.html#E5:29_1_1_2_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="closure2.html#K12">ClSys-&gt;ClOp</a> <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as  <a href="closure2.html#V7">reflexive</a> <a href="closure2.html#NM3">SetOp</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<a NAME="E8:29_1_1_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">c<sub>12</sub></font> <a href="closure2.html#K7">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="closure2.html#D12">CLOSURE2:def 12</a></i>;<br/>
<a NAME="E9:29_1_1_2_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="closure2.html#K7">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="closure3.html#E1:29_1_1_2_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E10:29_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="closure2.html#K7">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E6:29_1_1_2_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E11:29_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></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">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E4:29_1_1_2_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="closure3.html#E9:29_1_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E12:29_1_1_2_1"/>
<font color="Maroon">c<sub>10</sub></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">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="closure3.html#E1:29_1_1_2_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="closure3.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:29_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a>  <a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>


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

<b>thus </b><a NAME="E7:29_1_1"/>
 <a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:29_1_1_3"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:29_1_1_3_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:29_1_1_3_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:29_1_1_3_1_1"/>
<font color="Maroon">c<sub>10</sub></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">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E2:29_1_1_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></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">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E1:29_1_1_3_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="closure3.html#D4" target="_self">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:29_1_1_3_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E5:29_1_1_3_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> where B is    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></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">c<sub>12</sub></font> being    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E7:29_1_1_3_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E8:29_1_1_3_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="closure3.html#E5:29_1_1_3_1_1"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="closure3.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E10:29_1_1_3_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="closure2.html#K12">ClSys-&gt;ClOp</a> <span class="p2">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="closure2.html#K7">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E11:29_1_1_3_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> is <a href="pre_circ.html#V1">locally-finite</a> &amp;  <a href="closure3.html#K1" target="_self">supp</a> <font color="Maroon">c<sub>13</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>and </b><br/><a NAME="E12:29_1_1_3_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="closure3.html#E8:29_1_1_3_1_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> =  <a href="closure2.html#K12">ClSys-&gt;ClOp</a> <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as  <a href="closure2.html#V8">monotonic</a> <a href="closure2.html#NM3">SetOp</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure3.html#K5" target="_self">SubAlgCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<a NAME="E14:29_1_1_3_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="closure2.html#K7">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">c<sub>14</sub></font> <a href="closure2.html#K7">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="closure3.html#E12:29_1_1_3_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="closure2.html#D13">CLOSURE2:def 13</a></i>;<br/>
<a NAME="E15:29_1_1_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="closure3.html#E1:29_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="closure3.html#E1:29_1_1_3_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="closure3.html#E10:29_1_1_3_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<b>hence </b><a NAME="E16:29_1_1_3_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="closure3.html#E4:29_1_1_3_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="closure3.html#E7:29_1_1_3_1_1"><i><font color="Green">E15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:29_1_1_3"/>
 <a href="closure3.html#K2" target="_self">MSUnion</a> <font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>


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


</div>
