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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#NM1" title="PBOOLE:NM.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/><span class="kw">let </span><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#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>









<span class="kw">assume </span><a NAME="E1:116"/><span class="lab"><font color="Green" title="E67">A1</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A2</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T14" target="_self" title="PBOOLE:th.14">Th16</a></span>;<br/>
<a NAME="E3:116"/><span class="lab"><font color="Green" title="E69">A3</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="pboole.html#T14" target="_self" title="PBOOLE:th.14">Th16</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:116"/>
( <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E5:116"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A4</font></span>: 
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T111" target="_self" title="PBOOLE:th.111">Th121</a></span>;<br/>
<a NAME="E6:116"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="pboole.html#T14" target="_self" title="PBOOLE:th.14">Th16</a></span>;<br/>
<span class="kw">hence </span><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T23" target="_self" title="PBOOLE:th.23">Th25</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T32" target="_self" title="PBOOLE:th.32">Th38</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:116"><span class="lab"><font color="Green" title="E70">A4</font></span></a>, <a class="ref" href="pboole.html#T32" target="_self" title="PBOOLE:th.32">Th38</a></span>
<br/>.= 
<font color="Maroon" title="c3">V</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:116"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="pboole.html#T23" target="_self" title="PBOOLE:th.23">Th25</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<a NAME="E8:116"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="pboole.html#T14" target="_self" title="PBOOLE:th.14">Th16</a></span>;<br/>
<span class="kw">hence </span><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T23" target="_self" title="PBOOLE:th.23">Th25</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T32" target="_self" title="PBOOLE:th.32">Th38</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:116"><span class="lab"><font color="Green" title="E70">A4</font></span></a>, <a class="ref" href="pboole.html#T32" target="_self" title="PBOOLE:th.32">Th38</a></span>
<br/>.= 
<font color="Maroon" title="c2">Z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="ref" href="pboole.html#T23" target="_self" title="PBOOLE:th.23">Th25</a></span>
;<br/> <a class="txt" 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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