<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">I</font> be    <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>, <font color="Olive" title="b3">Z</font>, <font color="Olive" title="b4">V</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="b3">Z</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <font color="Olive" title="b4">V</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Olive" title="b1">X</font> <a href="pboole.html#R5" title="PBOOLE:pred.5">misses</a> <font color="Olive" title="b3">Z</font> &amp; <font color="Olive" title="b2">Y</font> <a href="pboole.html#R5" title="PBOOLE:pred.5">misses</a> <font color="Olive" title="b4">V</font> holds <br/>( <font color="Olive" title="b1">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b4">V</font> &amp; <font color="Olive" title="b2">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">Z</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</font>, <font color="Maroon" title="c4">Z</font>, <font color="Maroon" title="c5">V</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="c4">Z</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <font color="Maroon" title="c5">V</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <font color="Maroon" title="c3">Y</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">Z</font> &amp; <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R5" title="PBOOLE:pred.5">misses</a> <font color="Maroon" title="c5">V</font> implies ( <font color="Maroon" title="c2">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">V</font> &amp; <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">Z</font> ) )</span><br/>









<span class="kw">assume </span><a NAME="E1:110"/><span class="lab"><font color="Green" title="E61">A1</font></span>: 
<font color="Maroon" title="c4">Z</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <font color="Maroon" title="c5">V</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</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"> (  not <font color="Maroon" title="c2">X</font> <a href="pboole.html#R5" title="PBOOLE:pred.5">misses</a> <font color="Maroon" title="c4">Z</font> or  not <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R5" title="PBOOLE:pred.5">misses</a> <font color="Maroon" title="c5">V</font> or ( <font color="Maroon" title="c2">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">V</font> &amp; <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">Z</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:110"/>
( <font color="Maroon" title="c2">X</font> <a href="pboole.html#R5" title="PBOOLE:pred.5">misses</a> <font color="Maroon" title="c4">Z</font> &amp; <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R5" title="PBOOLE:pred.5">misses</a> <font color="Maroon" title="c5">V</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#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">V</font> &amp; <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">Z</font> )</span><br/>

<a NAME="E3:110"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A2</font></span>: 
( <font color="Maroon" title="c2">X</font> <a href="pboole.html#K3" title="PBOOLE:func.3">(/\)</a> <font color="Maroon" title="c4">Z</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> &amp; <font color="Maroon" title="c3">Y</font> <a href="pboole.html#K3" title="PBOOLE:func.3">(/\)</a> <font color="Maroon" title="c5">V</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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T111" target="_self" onmouseover="rs('pboole/T111')" onmouseout="rh()">Th111</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c2">X</font> = 
<font color="Maroon" title="c2">X</font> <a href="pboole.html#K3" title="PBOOLE:func.3">(/\)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Z</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <font color="Maroon" title="c5">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T23" target="_self" onmouseover="rs('pboole/T23')" onmouseout="rh()">Th23</a>, <a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="pboole.html#T14" target="_self" onmouseover="rs('pboole/T14')" onmouseout="rh()">Th14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K3" title="PBOOLE:func.3">(/\)</a> <font color="Maroon" title="c4">Z</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K3" title="PBOOLE:func.3">(/\)</a> <font color="Maroon" title="c5">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T32" target="_self" onmouseover="rs('pboole/T32')" onmouseout="rh()">Th32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">(/\)</a> <font color="Maroon" title="c5">V</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110"><span class="lab"><font color="Green" title="E62">A2</font></span></a>, <a class="ref" href="pboole.html#T32" target="_self" onmouseover="rs('pboole/T32')" onmouseout="rh()">Th32</a></span>
<br/>.= 
<font color="Maroon" title="c5">V</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="pboole.html#T14" target="_self" onmouseover="rs('pboole/T14')" onmouseout="rh()">Th14</a>, <a class="ref" href="pboole.html#T23" target="_self" onmouseover="rs('pboole/T23')" onmouseout="rh()">Th23</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"> <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">Z</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c3">Y</font> = 
<font color="Maroon" title="c3">Y</font> <a href="pboole.html#K3" title="PBOOLE:func.3">(/\)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Z</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <font color="Maroon" title="c5">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T23" target="_self" onmouseover="rs('pboole/T23')" onmouseout="rh()">Th23</a>, <a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="pboole.html#T14" target="_self" onmouseover="rs('pboole/T14')" onmouseout="rh()">Th14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="pboole.html#K3" title="PBOOLE:func.3">(/\)</a> <font color="Maroon" title="c4">Z</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="pboole.html#K3" title="PBOOLE:func.3">(/\)</a> <font color="Maroon" title="c5">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T32" target="_self" onmouseover="rs('pboole/T32')" onmouseout="rh()">Th32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="pboole.html#K3" title="PBOOLE:func.3">(/\)</a> <font color="Maroon" title="c4">Z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110"><span class="lab"><font color="Green" title="E62">A2</font></span></a>, <a class="ref" href="pboole.html#T32" target="_self" onmouseover="rs('pboole/T32')" onmouseout="rh()">Th32</a></span>
<br/>.= 
<font color="Maroon" title="c4">Z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="pboole.html#T14" target="_self" onmouseover="rs('pboole/T14')" onmouseout="rh()">Th14</a>, <a class="ref" href="pboole.html#T23" target="_self" onmouseover="rs('pboole/T23')" onmouseout="rh()">Th23</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/>


</div>
