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





<b>assume </b><a NAME="E1:74"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">X</font> is <a href="relat_1.html#V2">non-empty</a>
 ;<br/>

<b>thus </b><a NAME="E2:74"/>
<span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><span class="p2"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span>,<font color="Maroon">X</font></span><a href="pboole.html#K10">|]</a></span> is <a href="relat_1.html#V2">non-empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><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#D16">PBOOLE:def 16</a><br/>

<b>assume </b><a NAME="E1:74_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:74_1"/><b>then </b>
not <font color="Maroon">X</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/>
<a NAME="E3:74_1"/><b>then </b>
not <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<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="tarski.html#K1">}</a></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> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T130">ZFMISC_1:130</a></i>;<br/>
<a NAME="E4:74_1"/><b>then </b>
not <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <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> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:74_1"/>
not <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><span class="p2"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span>,<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> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>;<br/>


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

<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#D16">PBOOLE:def 16</a><br/>

<b>assume </b><a NAME="E3:74"/><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="E4:74"/><b>then </b>
not <font color="Maroon">X</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/>
<a NAME="E5:74"/><b>then </b>
not <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"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<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="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T130">ZFMISC_1:130</a></i>;<br/>
<a NAME="E6:74"/><b>then </b>
not <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"><span class="p3"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:74"/>
not <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">X</font>,<span class="p2"><a href="pzfmisc1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">x</font></span><a href="pzfmisc1.html#K1" target="_self">}</a></span></span><a href="pboole.html#K10">|]</a></span> <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#D21">PBOOLE:def 21</a></i>;<br/>


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