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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">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><a NAME="E1:77"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">X</font> <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 <font color="Maroon">X</font> <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="77_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1"/>
( <font color="Maroon">X</font> <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 <font color="Maroon">X</font> <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="77_1_1"><b>suppose </b></a><a NAME="E1:77_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">X</font> <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"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_1"><b>now </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="E1:77_1_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:77_1_1_1"/><b>then </b>
<font color="Maroon">X</font> <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="E3:77_1_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <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/><b>hence </b><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T135">ZFMISC_1:135</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pboole.html#K1">[0]</a> <font color="Maroon">I</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#T5">PBOOLE:5</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:77_1_1"/>
<font color="Maroon">X</font> <a href="pboole.html#R6">=</a>  <a href="pboole.html#K1">[0]</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2"><b>suppose </b></a><a NAME="E1:77_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">X</font> <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"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1"><b>now </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="E1:77_1_2_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:77_1_2_1"/><b>then </b>
<font color="Maroon">X</font> <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="E3:77_1_2_1"/><b>then </b>
<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <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/><b>hence </b><font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T135">ZFMISC_1:135</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pboole.html#K1">[0]</a> <font color="Maroon">I</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#T5">PBOOLE:5</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:77_1_2"/>
<font color="Maroon">X</font> <a href="pboole.html#R6">=</a>  <a href="pboole.html#K1">[0]</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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