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





<b>set </b><font color="Maroon" title="c4">AA</font> = <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c3">A</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">UA</font> =  <a href="mboolean.html#K2" title="MBOOLEAN:func.2">union</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c3">A</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1"/>
(  not <font color="Maroon" title="c3">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon" title="c3">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><b>suppose </b></a><a NAME="E1:16_1_1"/><i><font color="Green" title="E9">A1</font></i>: 
not <font color="Maroon" title="c3">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:16_1_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="c3">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">UA</font> <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="16_1_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="E1:16_1_1_1"/><i><font color="Green" title="E10">A2</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/>

<a NAME="E2:16_1_1_1"/><b>then </b><i><font color="Green" title="E11">A3</font></i>: 
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c3">A</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></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> <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="c3">A</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure3.html#E1:16_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="closure2.html#T15" title="CLOSURE2:th.15">CLOSURE2:15</a></i>;<br/>
<a NAME="E3:16_1_1_1"/>
<font color="Maroon" title="c5">UA</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>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><span class="p2"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c3">A</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="closure3.html#E1:16_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="mboolean.html#D2" title="MBOOLEAN:def.2">MBOOLEAN:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:16_1_1_1"/>
<span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c3">A</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> <font color="Maroon" title="c5">UA</font> <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#E1:16_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="closure3.html#E2:16_1_1_1"><i><font color="Green" title="E11">A3</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="E3:16_1_1"/>
 <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c3">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mboolean.html#K2" title="MBOOLEAN:func.2">union</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c3">A</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2"><b>suppose </b></a><a NAME="E1:16_1_2"/><i><font color="Green" title="E9">A4</font></i>: 
<font color="Maroon" title="c3">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font>
 ;<br/><div class="add"><a NAME="E2:16_1_2"/>
for <font color="Olive" title="b1">A</font> being <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font> holds   <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Olive" title="b1">A</font> is <a href="relat_1.html#V3" title="RELAT_1:attr.3">empty-yielding</a>
 ;<br/><a NAME="E3:16_1_2"/><b>then </b><i><font color="Green" title="E10">A5</font></i>: 
 <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c3">A</font> is <a href="relat_1.html#V3" title="RELAT_1:attr.3">empty-yielding</a>
 <b>by </b><i><a class="txt" href="closure3.html#E1:16_1_2"><i><font color="Green" title="E9">A4</font></i></a></i>;<br/><a NAME="E4:16_1_2"/>
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="c3">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">UA</font> <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="16_1_2_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="E1:16_1_2_1"/><i><font color="Green" title="E11">A6</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/>

<a NAME="E2:16_1_2_1"/>
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c3">A</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="txt" href="closure3.html#E1:16_1_2"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="closure2.html#D4" title="CLOSURE2:def.4">CLOSURE2:def 4</a></i>;<br/>
<a NAME="E3:16_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">UA</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="mboolean.html#T22" title="MBOOLEAN:th.22">MBOOLEAN:22</a></i>;<br/>
<a NAME="E4:16_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">UA</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="closure3.html#E1:16_1_2_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="pboole.html#D15" title="PBOOLE:def.15">PBOOLE:def 15</a></i>;<br/>
<b>hence </b><a NAME="E5:16_1_2_1"/>
<span class="p1">(<span class="default"><a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c3">A</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> <font color="Maroon" title="c5">UA</font> <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#E3:16_1_2"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="closure3.html#E1:16_1_2_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="pboole.html#D15" title="PBOOLE:def.15">PBOOLE:def 15</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:16_1_2"/>
 <a href="closure3.html#K2" title="CLOSURE3:func.2">MSUnion</a> <font color="Maroon" title="c3">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mboolean.html#K2" title="MBOOLEAN:func.2">union</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c3">A</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
