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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/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><a NAME="E2:39_1"/><b>then </b><i><font color="Green" title="E21">A2</font></i>: 
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><span class="p2">(<span class="default"><a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font></span>)</span></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="c3">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">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <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>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="closure2.html#T15" target="_self" title="CLOSURE2:th.15">Th15</a></i>;<br/><b>thus </b><a NAME="E3:39_1"/>
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><span class="p2">(<span class="default"><a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font></span>)</span></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="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:39_1_1"/>
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><span class="p2">(<span class="default"><a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font></span>)</span></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="c3">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_1_1_1"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><span class="p2">(<span class="default"><a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font></span>)</span></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="c3">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E3:39_1_1_1"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <b>and </b><br/><a NAME="E4:39_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="closure2.html#E2:39_1"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/>
<a NAME="E5:39_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E6:39_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 
<b>by </b><i><a class="txt" href="closure2.html#E1:39_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E7:39_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E8:39_1_1_1"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="closure2.html#E1:39_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="closure2.html#E3:39_1_1_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="mboolean.html#D1" title="MBOOLEAN:def.1">MBOOLEAN:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c4">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/>

<b>assume </b><a NAME="E2:39_1_1"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 ;<br/>

<a NAME="E3:39_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="txt" href="closure2.html#E1:39_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="pzfmisc1.html#T1" title="PZFMISC1:th.1">PZFMISC1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">X</font> = <span class="p1">(<span class="default"><a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c4">a</font></span>)</span> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:39_1_1"/><i><font color="Green" title="E23">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E6:39_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E24">A6</font></i>: <font color="Maroon" title="c5">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</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="c3">i</font>
<b>by </b><i><a class="txt" href="closure2.html#E5:39_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/>
<a NAME="E8:39_1_1"/>
<font color="Maroon" title="c5">X</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a>,<a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a><br/>

<b>assume </b><a NAME="E1:39_1_1_3"/><i><font color="Green" title="E25">A7</font></i>: 
<font color="Maroon" title="c6">s</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="hsNdiv(this)" href="javascript:()" title="39_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_3_1"/>
( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> or <font color="Maroon" title="c6">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_3_1_1"/><i><font color="Green" title="E26">A8</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font>
 ;<br/><div class="add"><a NAME="E2:39_1_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="closure2.html#E2:39_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="closure2.html#E1:39_1_1_3"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="mboolean.html#D1" title="MBOOLEAN:def.1">MBOOLEAN:def 1</a></i>;<br/><b>hence </b><a NAME="E3:39_1_1_3_1_1"/>
<font color="Maroon" title="c5">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="txt" href="closure2.html#E7:39_1_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="closure2.html#E1:39_1_1_3_1_1"><i><font color="Green" title="E26">A8</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_3_1_2"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font>
 ;<br/><div class="add"><a NAME="E2:39_1_1_3_1_2"/><b>then </b>
not <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="closure2.html#E5:39_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><font color="Maroon" title="c5">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> = 
<span class="p1">(<span class="default"><a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="pboole.html#T5" title="PBOOLE:th.5">PBOOLE:5</a></i>
;<br/><b>hence </b><a NAME="E4:39_1_1_3_1_2"/>
<font color="Maroon" title="c5">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E9:39_1_1"/><b>then </b>
<font color="Maroon" title="c5">X</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="closure2.html#D1" target="_self" title="CLOSURE2:def.1">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">x</font> being    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E11:39_1_1"/><i><font color="Green" title="E25">A9</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="closure2.html#E7:39_1_1"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/>
<b>thus </b><a NAME="E12:39_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><span class="p2">(<span class="default"><a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font></span>)</span></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="c3">i</font>
 <b>by </b><i><a class="txt" href="closure2.html#E2:39_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="closure2.html#E11:39_1_1"><i><font color="Green" title="E25">A9</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:39"/>
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><span class="p2">(<span class="default"><a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font></span>)</span></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>


</div>
