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<b>let </b><font color="Maroon" title="c1">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">X</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Olive" title="b2">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Olive" title="b1">X</font> <a href="pboole.html#NR6" title="PBOOLE:NR.6">does_not_overlap</a> <font color="Olive" title="b2">Y</font></span><br/><b>let </b><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c2">X</font> <a href="pboole.html#NR6" title="PBOOLE:NR.6">does_not_overlap</a> <font color="Maroon" title="c3">Y</font> )</span><br/>





<b>assume </b><a NAME="E1:129"/><i><font color="Green" title="E71">A1</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">X</font> <a href="pboole.html#NR6" title="PBOOLE:NR.6">does_not_overlap</a> <font color="Maroon" title="c3">Y</font></span><br/>

<b>assume </b><a NAME="E2:129"/>
<font color="Maroon" title="c2">X</font> <a href="pboole.html#R4" title="PBOOLE:pred.4">overlaps</a> <font color="Maroon" title="c3">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E4:129"/><i><font color="Green" title="E72">A2</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="pboole.html#R3" title="PBOOLE:pred.3">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">x</font> <a href="pboole.html#R3" title="PBOOLE:pred.3">in</a> <font color="Maroon" title="c3">Y</font> )
 <b>by </b><i><a class="ref" href="pboole.html#T11" target="_self" title="PBOOLE:th.11">Th11</a></i>;<br/>
<a NAME="E5:129"/>
<font color="Maroon" title="c4">x</font> <a href="pboole.html#R3" title="PBOOLE:pred.3">in</a> <font color="Maroon" title="c2">X</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c3">Y</font>
 
<b>by </b><i><a class="txt" href="#E4:129"><i><font color="Green" title="E72">A2</font></i></a>, <a class="ref" href="pboole.html#T8" target="_self" title="PBOOLE:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E6:129"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:129"><i><font color="Green" title="E71">A1</font></i></a>, <a class="ref" href="pboole.html#T136" target="_self" title="PBOOLE:th.136">Th136</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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