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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> ; <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">Z</font>, <font color="Olive" title="b2">V</font>, <font color="Olive" title="b3">X</font>, <font color="Olive" title="b4">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">Z</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Olive" title="b2">V</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Olive" title="b4">Y</font> &amp; <font color="Olive" title="b3">X</font> <a href="pboole.html#R5" title="PBOOLE:pred.5">misses</a> <font color="Olive" title="b1">Z</font> &amp; <font color="Olive" title="b4">Y</font> <a href="pboole.html#R5" title="PBOOLE:pred.5">misses</a> <font color="Olive" title="b2">V</font> holds <br/>( <font color="Olive" title="b3">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b4">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">Z</font> )</span><br/><b>let </b><font color="Maroon" title="c2">Z</font>, <font color="Maroon" title="c3">V</font>, <font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c5">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">Z</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">V</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c4">X</font> <a href="pboole.html#R5" title="PBOOLE:pred.5">misses</a> <font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="pboole.html#R5" title="PBOOLE:pred.5">misses</a> <font color="Maroon" title="c3">V</font> implies ( <font color="Maroon" title="c4">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">Z</font> ) )</span><br/>









<b>assume </b><a NAME="E1:116"/>
<font color="Maroon" title="c2">Z</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">V</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">X</font> <a href="pboole.html#R5" title="PBOOLE:pred.5">misses</a> <font color="Maroon" title="c2">Z</font> or  not <font color="Maroon" title="c5">Y</font> <a href="pboole.html#R5" title="PBOOLE:pred.5">misses</a> <font color="Maroon" title="c3">V</font> or ( <font color="Maroon" title="c4">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">Z</font> ) )</span><br/>

<a NAME="E2:116"/><b>then </b><i><font color="Green" title="E67">A1</font></i>: 
( <font color="Maroon" title="c4">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">Z</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">Z</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c2">Z</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c3">V</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">Y</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#T16" target="_self" title="PBOOLE:th.16">Th16</a></i>;<br/>
<b>assume </b><a NAME="E3:116"/>
( <font color="Maroon" title="c4">X</font> <a href="pboole.html#R5" title="PBOOLE:pred.5">misses</a> <font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="pboole.html#R5" title="PBOOLE:pred.5">misses</a> <font color="Maroon" title="c3">V</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">Z</font> )</span><br/>

<a NAME="E4:116"/><b>then </b><i><font color="Green" title="E68">A2</font></i>: 
( <font color="Maroon" title="c4">X</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c2">Z</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> &amp; <font color="Maroon" title="c5">Y</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c3">V</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> )
 
<b>by </b><i><a class="ref" href="pboole.html#T121" target="_self" title="PBOOLE:th.121">Th121</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">X</font> = 
<font color="Maroon" title="c4">X</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">V</font></span>)</span>
<b>by </b><i><a class="txt" href="pboole.html#E2:116"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="pboole.html#T25" target="_self" title="PBOOLE:th.25">Th25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c3">V</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#T38" target="_self" title="PBOOLE:th.38">Th38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">Y</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c3">V</font>
<b>by </b><i><a class="txt" href="pboole.html#E4:116"><i><font color="Green" title="E68">A2</font></i></a>, <a class="ref" href="pboole.html#T38" target="_self" title="PBOOLE:th.38">Th38</a></i>
<br/>.= 
<font color="Maroon" title="c3">V</font>
<b>by </b><i><a class="txt" href="pboole.html#E2:116"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="pboole.html#T25" target="_self" title="PBOOLE:th.25">Th25</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">Z</font></span><br/>

<b>thus </b><font color="Maroon" title="c5">Y</font> = 
<font color="Maroon" title="c5">Y</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Z</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">V</font></span>)</span>
<b>by </b><i><a class="txt" href="pboole.html#E2:116"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="pboole.html#T25" target="_self" title="PBOOLE:th.25">Th25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c2">Z</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c3">V</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#T38" target="_self" title="PBOOLE:th.38">Th38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">Y</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c2">Z</font>
<b>by </b><i><a class="txt" href="pboole.html#E4:116"><i><font color="Green" title="E68">A2</font></i></a>, <a class="ref" href="pboole.html#T38" target="_self" title="PBOOLE:th.38">Th38</a></i>
<br/>.= 
<font color="Maroon" title="c2">Z</font>
<b>by </b><i><a class="txt" href="pboole.html#E2:116"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="pboole.html#T25" target="_self" title="PBOOLE:th.25">Th25</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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