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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1_1_1"/>
(  not <font color="Maroon" title="c3">SF</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon" title="c3">SF</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_1_1_1"/><i><font color="Green" title="E16">A1</font></i>: 
not <font color="Maroon" title="c3">SF</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:34_1_1_1_1"/>
<span class="p1"><a href="closure2.html#K4" title="CLOSURE2:func.4">|:</a><span class="default"><font color="Maroon" title="c3">SF</font></span><a href="closure2.html#K4" title="CLOSURE2:func.4">:|</a></span> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</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="E1:34_1_1_1_1_1"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<a NAME="E2:34_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E18">A3</font></i>: 
<span class="p1"><a href="closure2.html#K4" title="CLOSURE2:func.4">|:</a><span class="default"><font color="Maroon" title="c3">SF</font></span><a href="closure2.html#K4" title="CLOSURE2:func.4">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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="c4">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> <font color="Maroon" title="c3">SF</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure2.html#E1:34_1_1_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="closure2.html#T15" target="_self" title="CLOSURE2:th.15">Th15</a></i>;<br/>
<b>thus </b><a NAME="E3:34_1_1_1_1_1"/>
<span class="p1"><a href="closure2.html#K4" title="CLOSURE2:func.4">|:</a><span class="default"><font color="Maroon" title="c3">SF</font></span><a href="closure2.html#K4" title="CLOSURE2:func.4">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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="c4">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</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="E1:34_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="closure2.html#K4" title="CLOSURE2:func.4">|:</a><span class="default"><font color="Maroon" title="c3">SF</font></span><a href="closure2.html#K4" title="CLOSURE2:func.4">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">a</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="E3:34_1_1_1_1_1_1"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>and </b><br/><a NAME="E4:34_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">SF</font>
 <b>by </b><i><a class="txt" href="closure2.html#E2:34_1_1_1_1_1"><i><font color="Green" title="E18">A3</font></i></a></i>;<br/>
<a NAME="E5:34_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">a</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="E6:34_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</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="c4">i</font>
 
<b>by </b><i><a class="txt" href="closure2.html#E1:34_1_1_1_1_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="closure2.html#E3:34_1_1_1_1_1_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E7:34_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</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="c4">i</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E8:34_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">x</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="c4">i</font>
 <b>by </b><i><a class="txt" href="closure2.html#E1:34_1_1_1_1_1"><i><font color="Green" title="E17">A2</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>


</div><b>end;</b></div><b>hence </b><a NAME="E3:34_1_1_1_1"/>
<span class="p1"><a href="closure2.html#K4" title="CLOSURE2:func.4">|:</a><span class="default"><font color="Maroon" title="c3">SF</font></span><a href="closure2.html#K4" title="CLOSURE2:func.4">:|</a></span> is   <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_1_1_2"/>
<font color="Maroon" title="c3">SF</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:34_1_1_1_2"/><b>then </b>
<span class="p1"><a href="closure2.html#K4" title="CLOSURE2:func.4">|:</a><span class="default"><font color="Maroon" title="c3">SF</font></span><a href="closure2.html#K4" title="CLOSURE2:func.4">:|</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="ref" href="closure2.html#D4" target="_self" title="CLOSURE2:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E3:34_1_1_1_2"/>
<span class="p1"><a href="closure2.html#K4" title="CLOSURE2:func.4">|:</a><span class="default"><font color="Maroon" title="c3">SF</font></span><a href="closure2.html#K4" title="CLOSURE2:func.4">:|</a></span> is   <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="mssubfam.html#T31" title="MSSUBFAM:th.31">MSSUBFAM:31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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