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<b>let </b><font color="Maroon">I</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">M</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">D</font> be  <a href="mssubfam.html#V5">properly-upper-bound</a> <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font>;<br/>







<a NAME="E1:70"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">D</font> <a href="pboole.html#R2">c=</a>  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">SF</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:70"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/>for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> iff ( <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">e</font>] ) )
 <b>from </b><i><a class="ref" href="pboole.html#S2">PBOOLE:sch 2</a>();<br/></i>
<a NAME="E4:70"/>
<font color="Maroon">SF</font> is    <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:70_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:70_1"/><b>then </b>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E3:70_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mboolean.html#D1">MBOOLEAN:def 1</a></i>;<br/>
<b>thus </b><a NAME="E4:70_1"/>
<font color="Maroon">SF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:70_1_1"/>
<font color="Maroon">SF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:70_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E2:70_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:70_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:70_1_1"/>
<font color="Maroon">SF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="mboolean.html#D1">MBOOLEAN:def 1</a></i>;<br/>


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


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">SF</font> = <font color="Maroon">SF</font> as    <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font> ;<br/>
<b>reconsider </b><font color="Maroon">SF</font> = <font color="Maroon">SF</font> as   <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">M</font> ;<br/>
<a NAME="E7:70"/>
<font color="Maroon">SF</font> is <a href="relat_1.html#V2">non-empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a><br/>

<b>assume </b><a NAME="E1:70_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:70_2"/>
<font color="Maroon">X</font> <a href="pboole.html#R1">in</a>  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T12">MSSUBFAM:12</a></i>;<br/>
<a NAME="E3:70_2"/><b>then </b>
<font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="mboolean.html#T19">MBOOLEAN:19</a></i>;<br/>
<a NAME="E4:70_2"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E5:70_2"/>
<font color="Maroon">M</font> <a href="pboole.html#R1">in</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#D7">MSSUBFAM:def 7</a></i>;<br/>
<a NAME="E6:70_2"/><b>then </b>
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:70_2"/>
not <font color="Maroon">SF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">SF</font> = <font color="Maroon">SF</font> as  <a href="relat_1.html#V2">V5</a> <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">M</font> ;<br/>
<b>take </b>
<font color="Maroon">SF</font>
;<br/>

<b>let </b><font color="Maroon">Y</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>

<b>thus </b><a NAME="E9:70"/>
( <font color="Maroon">Y</font> <a href="pboole.html#R1">in</a> <font color="Maroon">SF</font> implies ( <font color="Maroon">Y</font> <a href="pboole.html#R1">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">Y</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">Y</font> <a href="pboole.html#R1">in</a> <font color="Maroon">SF</font>
 ;<br/>

<b>thus </b><a NAME="E2:70_3"/>
<font color="Maroon">Y</font> <a href="pboole.html#R1">in</a> <font color="Maroon">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D4">PBOOLE:def 4</a><br/>

<b>assume </b><a NAME="E1:70_3_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:70_3_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:70_3_1"/>
<font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E3:70_3"/>
<font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:70_3_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:70_3_2"/><b>then </b>
<font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:70_3_2"/>
<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/>


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


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

<b>assume </b><a NAME="E10:70"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">Y</font> <a href="pboole.html#R1">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">Y</font> )
 ;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D4">PBOOLE:def 4</a><br/>

<b>assume </b><a NAME="E11:70"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E12:70"/><b>then </b>
( <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i>, <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<b>hence </b><a NAME="E13:70"/>
<font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/>


</div>
