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<div class="add">

<b>let </b><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">M</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/><b>let </b><font color="Maroon" title="c3">AA</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E8">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">AA</font> holds <br/><font color="Olive" title="b1">x</font> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font> be   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:21"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">AA</font> </span>}</span> 
 <b>and </b><br/><a NAME="E3:21"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">AA</font>
 ;<br/>

<b>thus </b><a NAME="E4:21"/>
 <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</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="c4">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="21_1"><b>proof </b></a><div class="add">

<a NAME="E1:21_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:21_1_1"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:21_1_1_1"/>
( <font color="Maroon" title="c7">a</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="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> implies <font color="Maroon" title="c7">a</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="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1_1_1"/>
<font color="Maroon" title="c7">a</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="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 ;<br/>

<a NAME="E2:21_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">a</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">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">f</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">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E1:21_1_1"><i><font color="Green" title="E11">A4</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="c8">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:21_1_1_1_1"/><i><font color="Green" title="E12">A5</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 <b>and </b><br/><a NAME="E5:21_1_1_1_1"/><i><font color="Green" title="E13">A6</font></i>: 
<font color="Maroon" title="c8">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">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">f</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">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</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="c9">f</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="E7:21_1_1_1_1"/><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Y</font>
 <b>and </b><br/><a NAME="E8:21_1_1_1_1"/><i><font color="Green" title="E15">A8</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="txt" href="closure3.html#E5:21_1_1_1_1"><i><font color="Green" title="E13">A6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">Q</font> being   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E10:21_1_1_1_1"/><i><font color="Green" title="E16">A9</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c10">Q</font>
 <b>and </b><br/><a NAME="E11:21_1_1_1_1"/><i><font color="Green" title="E17">A10</font></i>: 
<font color="Maroon" title="c10">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">AA</font>
 <b>by </b><i><a class="txt" href="closure3.html#E2:21"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="closure3.html#E8:21_1_1_1_1"><i><font color="Green" title="E15">A8</font></i></a></i>;<br/>
<a NAME="E12:21_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c10">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c6">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="c10">Q</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E1:21_1_1"><i><font color="Green" title="E11">A4</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="c11">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E14:21_1_1_1_1"/><i><font color="Green" title="E18">A11</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">d</font>
 <b>and </b><br/><a NAME="E15:21_1_1_1_1"/><i><font color="Green" title="E19">A12</font></i>: 
<font color="Maroon" title="c11">d</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="c6">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="c10">Q</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="closure3.html#E4:21_1_1_1_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="closure3.html#E7:21_1_1_1_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="closure3.html#E10:21_1_1_1_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">g</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="E17:21_1_1_1_1"/><i><font color="Green" title="E20">A13</font></i>: 
<font color="Maroon" title="c11">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <b>and </b><br/><a NAME="E18:21_1_1_1_1"/><i><font color="Green" title="E21">A14</font></i>: 
<font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q</font>
 <b>by </b><i><a class="txt" href="closure3.html#E15:21_1_1_1_1"><i><font color="Green" title="E19">A12</font></i></a></i>;<br/>
<a NAME="E19:21_1_1_1_1"/>
<font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">AA</font>
 
<b>by </b><i><a class="txt" href="closure3.html#E11:21_1_1_1_1"><i><font color="Green" title="E17">A10</font></i></a>, <a class="txt" href="closure3.html#E18:21_1_1_1_1"><i><font color="Green" title="E21">A14</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E20:21_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">d</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">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">h</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">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">AA</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E17:21_1_1_1_1"><i><font color="Green" title="E20">A13</font></i></a></i>;<br/>
<a NAME="E21:21_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">a</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">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">h</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">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">AA</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E14:21_1_1_1_1"><i><font color="Green" title="E18">A11</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="E22:21_1_1_1_1"/>
<font color="Maroon" title="c7">a</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="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="closure3.html#E3:21"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="closure3.html#E1:21_1_1"><i><font color="Green" title="E11">A4</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></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_1_1"/>
<span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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>
<b>hence </b><a NAME="E2:21_1"/>
 <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</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="c4">A</font>
 <b>by </b><i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>


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

<a NAME="E5:21"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:21_2"/><i><font color="Green" title="E11">A15</font></i>: 
<font color="Maroon" title="c6">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="c7">a</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/>

<a NAME="E2:21_2"/>
( <font color="Maroon" title="c7">a</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="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> implies <font color="Maroon" title="c7">a</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="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2_1"/>
<font color="Maroon" title="c7">a</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="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 ;<br/>

<a NAME="E2:21_2_1"/><b>then </b>
<font color="Maroon" title="c7">a</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">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">f</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">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E1:21_2"><i><font color="Green" title="E11">A15</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="c8">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:21_2_1"/><i><font color="Green" title="E12">A16</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 <b>and </b><br/><a NAME="E5:21_2_1"/><i><font color="Green" title="E13">A17</font></i>: 
<font color="Maroon" title="c8">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">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">f</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">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">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="c9">f</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="E7:21_2_1"/><i><font color="Green" title="E14">A18</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Y</font>
 <b>and </b><br/><a NAME="E8:21_2_1"/><i><font color="Green" title="E15">A19</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="closure3.html#E5:21_2_1"><i><font color="Green" title="E13">A17</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:21_2_1"/><i><font color="Green" title="E16">A20</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">c</font>
 <b>and </b><br/><a NAME="E11:21_2_1"/><i><font color="Green" title="E17">A21</font></i>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">AA</font>
 <b>by </b><i><a class="txt" href="closure3.html#E3:21"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="closure3.html#E8:21_2_1"><i><font color="Green" title="E15">A19</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">c</font> = <font color="Maroon" title="c10">c</font> as   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font> <b>by </b><i><a class="txt" href="closure3.html#E1:21"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="closure3.html#E11:21_2_1"><i><font color="Green" title="E17">A21</font></i></a></i>;<br/>
<a NAME="E13:21_2_1"/><i><font color="Green" title="E18">A22</font></i>: 
<span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">v</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">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">c</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E1:21_2"><i><font color="Green" title="E11">A15</font></i></a>, <a class="ref" href="closure3.html#D4" target="_self" title="CLOSURE3:def.4">Def4</a></i>;<br/>
<a NAME="E14:21_2_1"/>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">v</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">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">c</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E10:21_2_1"><i><font color="Green" title="E16">A20</font></i></a></i>;<br/>
<a NAME="E15:21_2_1"/><b>then </b><i><font color="Green" title="E19">A23</font></i>: 
<font color="Maroon" title="c7">a</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">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">v</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">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">c</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E4:21_2_1"><i><font color="Green" title="E12">A16</font></i></a>, <a class="txt" href="closure3.html#E7:21_2_1"><i><font color="Green" title="E14">A18</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E16:21_2_1"/><i><font color="Green" title="E20">A24</font></i>: 
 <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Olive" title="b1">X</font></span>)</span> where <font color="Olive" title="b1">X</font> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">AA</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E11:21_2_1"><i><font color="Green" title="E17">A21</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">cos</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E18:21_2_1"/><i><font color="Green" title="E21">A25</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">cos</font>
 <b>and </b><br/><a NAME="E19:21_2_1"/><i><font color="Green" title="E22">A26</font></i>: 
<font color="Maroon" title="c12">cos</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c11">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="closure3.html#E13:21_2_1"><i><font color="Green" title="E18">A22</font></i></a>, <a class="txt" href="closure3.html#E15:21_2_1"><i><font color="Green" title="E19">A23</font></i></a></i>;<br/>
<a NAME="E20:21_2_1"/>
<font color="Maroon" title="c12">cos</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">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">t</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">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E2:21"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="closure3.html#E16:21_2_1"><i><font color="Green" title="E20">A24</font></i></a>, <a class="txt" href="closure3.html#E19:21_2_1"><i><font color="Green" title="E22">A26</font></i></a></i>;<br/>
<a NAME="E21:21_2_1"/><b>then </b>
<font color="Maroon" title="c7">a</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">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> where <font color="Olive" title="b1">t</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">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E18:21_2_1"><i><font color="Green" title="E21">A25</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="E22:21_2_1"/>
<font color="Maroon" title="c7">a</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="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="closure3.html#E1:21_2"><i><font color="Green" title="E11">A15</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>hence </b><a NAME="E3:21_2"/>
(  not <font color="Maroon" title="c7">a</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="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> or <font color="Maroon" title="c7">a</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="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:21"/>
 <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c4">A</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="c5">B</font>
 <b>by </b><i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>


</div>
