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<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">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> where <font color="Olive">f</font> is    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> ;<br/>
<a NAME="E1:13_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<a NAME="E2:13_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon">IT</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:13_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E5:13_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<b>reconsider </b><font color="Maroon">IT</font> = <font color="Maroon">IT</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:13_1_1"/>
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/><font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:13_1_1_1"/><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="E2:13_1_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_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> ;<br/>

<b>assume </b><a NAME="E1:13_1_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E2:13_1_1_1_1"/>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">f</font> is    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:13_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E5:13_1_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">f</font> is    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">ff</font> being    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font><b> such that </b><br/><a NAME="E7:13_1_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E8:13_1_1_1_1"/>
<font color="Maroon">ff</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">ff</font> as    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">M</font> ;<br/>
<a NAME="E10:13_1_1_1_1"/>
<font color="Maroon">ff</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E11:13_1_1_1_1"/><b>then </b>
<font color="Maroon">ff</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/>
<b>hence </b><a NAME="E12:13_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</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>hence </b><a NAME="E3:13_1_1_1"/>
<font color="Maroon">IT</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="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:13_1_1"/><b>then </b>
<font color="Maroon">IT</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">IT</font> = <font color="Maroon">IT</font> as    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<b>take </b>
<font color="Maroon">IT</font>
;<br/>

<b>thus </b><a NAME="E10:13_1_1"/>
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/><font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">f</font> is    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <font color="Maroon">M</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 <b>by </b><i/>;<br/>


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