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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">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>thus </b><a NAME="E1:20"/>
( <font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">A</font> <a href="pboole.html#K5">\+\</a> <font color="Maroon">B</font> implies ( <font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">A</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">B</font> &amp; <font color="Maroon">X</font> <a href="pboole.html#R5">misses</a> <font color="Maroon">A</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">B</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1"/>
<font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">A</font> <a href="pboole.html#K5">\+\</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:20_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">X</font> <a href="pboole.html#R1">in</a>  <a href="mboolean.html#K1" target="_self">bool</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pboole.html#K5">\+\</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E3:20_1"/>
<font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">A</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_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> ; <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="E1:20_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:20_1_1"/><b>then </b>
<font color="Maroon">X</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#K1" target="_self">bool</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pboole.html#K5">\+\</a> <font color="Maroon">B</font></span>)</span></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="E3:20_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<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> <a href="xboole_0.html#K5">\+\</a> <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>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:20_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">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T107">XBOOLE_1:107</a></i>;<br/>
<b>hence </b><a NAME="E5:20_1_1"/>
<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">(<span class="default"><font color="Maroon">A</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">B</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#D7">PBOOLE:def 7</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#D11">PBOOLE:def 11</a><br/>

<b>assume </b><a NAME="E4:20_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="E5:20_1"/><b>then </b>
<font color="Maroon">X</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#K1" target="_self">bool</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pboole.html#K5">\+\</a> <font color="Maroon">B</font></span>)</span></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="E6:20_1"/><b>then </b>
<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<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> <a href="xboole_0.html#K5">\+\</a> <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>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:20_1"/><b>then </b>
<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T107">XBOOLE_1:107</a></i>;<br/>
<b>hence </b><a NAME="E8:20_1"/>
<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">B</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#D8">PBOOLE:def 8</a></i>;<br/>


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

<b>assume </b><a NAME="E2:20"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">A</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">B</font> &amp; <font color="Maroon">X</font> <a href="pboole.html#R5">misses</a> <font color="Maroon">A</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">B</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#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E3:20"/><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="E4:20"/><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">(<span class="default"><font color="Maroon">A</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">B</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#D5">PBOOLE:def 5</a>, <a class="ref" href="pboole.html#D11">PBOOLE:def 11</a></i>;<br/>
<a NAME="E5:20"/><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">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="pboole.html#D7">PBOOLE:def 7</a>, <a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>;<br/>
<a NAME="E6:20"/><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">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K5">\+\</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T107">XBOOLE_1:107</a></i>;<br/>
<b>hence </b><a NAME="E7:20"/>
<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">(<span class="default"><font color="Maroon">A</font> <a href="pboole.html#K5">\+\</a> <font color="Maroon">B</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#T4">PBOOLE:4</a></i>;<br/>


</div>
