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









<b>assume </b><b>that </b><br/><a NAME="E1:82"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">A</font> is <a href="relat_1.html#V2">non-empty</a>
 <b>and </b><br/><a NAME="E2:82"/><i><font color="Green">E27</font></i>: 
( <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#R2">c=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="pboole.html#K10">|]</a></span> or <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#R2">c=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="pboole.html#K10">|]</a></span> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_1"/>
( <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#R2">c=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="pboole.html#K10">|]</a></span> or <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#R2">c=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="pboole.html#K10">|]</a></span> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1"><b>suppose </b></a><a NAME="E1:82_1_1"/><i><font color="Green">E28</font></i>: 
<span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#R2">c=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="pboole.html#K10">|]</a></span>
 ;<br/><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="E2:82_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E3:82_1_1"/><b>then </b>
<span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="pboole.html#K10">|]</a></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="E4:82_1_1"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>;<br/><a NAME="E5:82_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>;<br/><a NAME="E6:82_1_1"/>
not <font color="Maroon">A</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>, , <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/><b>hence </b><a NAME="E7:82_1_1"/>
<font color="Maroon">B</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="zfmisc_1.html#T139">ZFMISC_1:139</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2"><b>suppose </b></a><a NAME="E1:82_1_2"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#R2">c=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="pboole.html#K10">|]</a></span>
 ;<br/><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="E2:82_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E3:82_1_2"/><b>then </b>
<span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="pboole.html#K10">|]</a></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="E4:82_1_2"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>;<br/><a NAME="E5:82_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>;<br/><a NAME="E6:82_1_2"/>
not <font color="Maroon">A</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>, , <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/><b>hence </b><a NAME="E7:82_1_2"/>
<font color="Maroon">B</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="zfmisc_1.html#T139">ZFMISC_1:139</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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