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<span class="kw">let </span><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#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.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">EmptyMS</a> <font color="Maroon" title="c1">I</font> holds <br/> not <font color="Olive" title="b1">X</font> <a href="pboole.html#R4" title="PBOOLE:pred.4">overlaps</a> <font color="Olive" title="b2">Y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">EmptyMS</a> <font color="Maroon" title="c1">I</font> implies  not <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> )</span><br/>





<span class="kw">assume </span><a NAME="E1:120"/><span class="lab"><font color="Green" title="E64">A1</font></span>: 
<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">EmptyMS</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <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></span><br/>

<span class="kw">assume </span><a NAME="E2:120"/>
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><span class="kw"> such that </span><br/><a NAME="E4:120"/><span class="lab"><font color="Green" title="E65">A2</font></span>: 
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T11" target="_self" onmouseover="rs('pboole/T11')" onmouseout="rh()">Th11</a></span>;<br/>
<a NAME="E5:120"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:120"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="ref" href="pboole.html#T8" target="_self" onmouseover="rs('pboole/T8')" onmouseout="rh()">Th8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:120"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:120"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="ref" href="pboole.html#T124" target="_self" onmouseover="rs('pboole/T124')" onmouseout="rh()">Th124</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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