<?xml version="1.0"?>
<div class="add">

<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">SF</font> be   <a href="closure2.html#NM1">SubsetFamily</a> of <font color="Maroon">M</font>;<br/>





<b>assume </b><a NAME="E1:26"/><i><font color="Green">E36</font></i>: 
not <font color="Maroon">SF</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:26"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">SF</font>
 <b>and </b><br/><a NAME="E4:26"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="closure2.html#K3" target="_self">|.</a><span class="default"><font color="Maroon">SF</font></span><a href="closure2.html#K3" target="_self">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E5:26"/><i><font color="Green">E40</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>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="closure2.html#K3" target="_self">|.</a><span class="default"><font color="Maroon">SF</font></span><a href="closure2.html#K3" target="_self">.|</a></span> holds <br/><span class="p1"><a href="closure2.html#K3" target="_self">|.</a><span class="default"><font color="Maroon">SF</font></span><a href="closure2.html#K3" target="_self">.|</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : verum </span>}</span> 
 <b>by </b><i/>;<br/>
<a NAME="E6:26"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="closure2.html#K4" target="_self">|:</a><span class="default"><font color="Maroon">SF</font></span><a href="closure2.html#K4" target="_self">:|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="closure2.html#K3" target="_self">|.</a><span class="default"><font color="Maroon">SF</font></span><a href="closure2.html#K3" target="_self">.|</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E7:26"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E8:26"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="closure2.html#K4" target="_self">|:</a><span class="default"><font color="Maroon">SF</font></span><a href="closure2.html#K4" target="_self">:|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">K</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">x</font> is    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">N</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : verum </span>}</span> ;<br/>
<a NAME="E9:26"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">x</font> is    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:26_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">x</font> is    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : verum </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:26_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">x</font> is    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font><b> such that </b><br/><a NAME="E3:26_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> )
 ;<br/>
<b>thus </b><a NAME="E4:26_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : verum </span>}</span> 
 <b>by </b><i>, </i>;<br/>


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

<b>let </b><font color="Maroon">q</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="E2:26_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:26_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>
<a NAME="E5:26_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:26_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">x</font> is    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:26"/>
<span class="p1"><a href="closure2.html#K4" target="_self">|:</a><span class="default"><font color="Maroon">SF</font></span><a href="closure2.html#K4" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">x</font> is    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> </span>}</span> 
 <b>by </b><i>, , , </i>;<br/>


</div>
