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<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> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <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> iff <font color="Olive" title="b1">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Olive" title="b2">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">X</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> iff <font color="Maroon" title="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">X</font> )</span><br/>





<div><span class="kw">hereby </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="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">X</font> implies <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> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:71_1"/><span class="lab"><font color="Green" title="E44">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"> <font color="Maroon" title="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font></span><br/><span class="kw">thus </span><font color="Maroon" title="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Maroon" title="c3">Y</font> = 
<font color="Maroon" title="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</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="c3">Y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T70" target="_self" onmouseover="rs('pboole/T70')" onmouseout="rh()">Th70</a></span>
<br/>.= 
<font color="Maroon" title="c2">X</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="pboole.html#T59" target="_self" onmouseover="rs('pboole/T59')" onmouseout="rh()">Th59</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>
<span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:71"/>
( <font color="Maroon" title="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">X</font> implies <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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T63" target="_self" onmouseover="rs('pboole/T63')" onmouseout="rh()">Th63</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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