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<b>let </b><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/>







<b>thus </b><a NAME="E1:18"/>
( <font color="Maroon" title="c2">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">A</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c4">B</font> implies ( <font color="Maroon" title="c2">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c2">X</font> <a href="pboole.html#R5" title="PBOOLE:pred.5">misses</a> <font color="Maroon" title="c4">B</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon" title="c2">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">A</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c4">B</font>
 ;<br/>

<a NAME="E2:18_1"/><b>then </b><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mboolean.html#K1" title="MBOOLEAN:func.1">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mboolean.html#T1" target="_self" title="MBOOLEAN:th.1">Th1</a></i>;<br/>
<b>thus </b><a NAME="E3:18_1"/>
<font color="Maroon" title="c2">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:18_1_1"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<a NAME="E2:18_1_1"/><b>then </b>
<font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="mboolean.html#K1" title="MBOOLEAN:func.1">bool</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 
<b>by </b><i><a class="txt" href="mboolean.html#E2:18_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="pboole.html#D4" title="PBOOLE:def.4">PBOOLE:def 4</a></i>;<br/>
<a NAME="E3:18_1_1"/><b>then </b>
<font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mboolean.html#E1:18_1_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="mboolean.html#E5"><i><font color="Green" title="E5">Lm4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:18_1_1"/>
<font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D11" title="PBOOLE:def.11">PBOOLE:def 11</a><br/>

<b>assume </b><a NAME="E4:18_1"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<a NAME="E5:18_1"/><b>then </b>
<font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="mboolean.html#K1" title="MBOOLEAN:func.1">bool</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 
<b>by </b><i><a class="txt" href="mboolean.html#E2:18_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="pboole.html#D4" title="PBOOLE:def.4">PBOOLE:def 4</a></i>;<br/>
<a NAME="E6:18_1"/><b>then </b>
<font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mboolean.html#E4:18_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="mboolean.html#E5"><i><font color="Green" title="E5">Lm4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:18_1"/>
<font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T106" title="XBOOLE_1:th.106">XBOOLE_1:106</a></i>;<br/>


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

<b>assume </b><a NAME="E2:18"/><i><font color="Green" title="E8">A4</font></i>: 
( <font color="Maroon" title="c2">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c2">X</font> <a href="pboole.html#R5" title="PBOOLE:pred.5">misses</a> <font color="Maroon" title="c4">B</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E3:18"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<a NAME="E4:18"/><b>then </b>
( <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )
 
<b>by </b><i><a class="txt" href="mboolean.html#E2:18"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a>, <a class="ref" href="pboole.html#D11" title="PBOOLE:def.11">PBOOLE:def 11</a></i>;<br/>
<a NAME="E5:18"/><b>then </b>
<font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T86" title="XBOOLE_1:th.86">XBOOLE_1:86</a></i>;<br/>
<b>hence </b><a NAME="E6:18"/>
<font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="mboolean.html#E3:18"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="pboole.html#D9" title="PBOOLE:def.9">PBOOLE:def 9</a></i>;<br/>


</div>
