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





<b>thus </b><a NAME="E1:31"/>
( <font color="Maroon">A</font> <a href="pboole.html#R1">in</a>  <a href="mboolean.html#K2" target="_self">union</a> <font color="Maroon">X</font> implies  ex <font color="Olive">Y</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> st <br/>( <font color="Maroon">A</font> <a href="pboole.html#R1">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> <a href="pboole.html#R1">in</a> <font color="Maroon">X</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">A</font> <a href="pboole.html#R1">in</a>  <a href="mboolean.html#K2" target="_self">union</a> <font color="Maroon">X</font>
 ;<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">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E2:31_1"/><i><font color="Green">E23</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/> 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">i</font>,<font color="Olive">Y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_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="E1:31_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:31_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="mboolean.html#K2" target="_self">union</a> <font color="Maroon">X</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#D4">PBOOLE:def 4</a></i>;<br/>
<a NAME="E3:31_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:31_1_1"/>
 ex <font color="Olive">Y'</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y'</font> &amp; <font color="Olive">Y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">K</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E4:31_1"/><i><font color="Green">E32</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/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">K</font>
;<br/>

<b>thus </b><a NAME="E5:31_1"/>
<font color="Maroon">A</font> <a href="pboole.html#R1">in</a> <font color="Maroon">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_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#D4">PBOOLE:def 4</a><br/>

<b>assume </b><a NAME="E1:31_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>hence </b><a NAME="E2:31_1_2"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E6:31_1"/>
<font color="Maroon">K</font> <a href="pboole.html#R1">in</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_3"><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="E1:31_1_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>hence </b><a NAME="E2:31_1_3"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/>


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


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

<b>given </b><font color="Maroon">Y</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><a NAME="E2:31"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">A</font> <a href="pboole.html#R1">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="pboole.html#R1">in</a> <font color="Maroon">X</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="E3:31"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E4:31"/><b>then </b>
( <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <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">X</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/>
<a NAME="E5:31"/><b>then </b>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:31"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="mboolean.html#K2" target="_self">union</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/>


</div>
