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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>, <font color="Olive" title="b3">Z</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#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b2">Y</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <font color="Olive" title="b3">Z</font> holds <br/>( <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#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b3">Z</font> &amp; <font color="Olive" title="b1">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Olive" title="b3">Z</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</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>, <font color="Maroon" title="c4">Z</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#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">Y</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <font color="Maroon" title="c4">Z</font> implies ( <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#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">Z</font> &amp; <font color="Maroon" title="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Maroon" title="c4">Z</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">Y</font> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:76"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">Y</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <font color="Maroon" title="c4">Z</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="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">Z</font> &amp; <font color="Maroon" title="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Maroon" title="c4">Z</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">Y</font> )</span><br/>

<a NAME="E2:76"/><span class="kw">then </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> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</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="c3">Y</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T53" target="_self" onmouseover="rs('pboole/T53')" onmouseout="rh()">Th53</a></span>;<br/>
<a NAME="E3:76"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A2</font></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> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">Z</font> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T75" target="_self" onmouseover="rs('pboole/T75')" onmouseout="rh()">Th75</a></span>;<br/>
<a NAME="E4:76"/>
<font color="Maroon" title="c4">Z</font> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Maroon" title="c3">Y</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T56" target="_self" onmouseover="rs('pboole/T56')" onmouseout="rh()">Th56</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:76"/>
<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#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:76"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="pboole.html#T13" target="_self" onmouseover="rs('pboole/T13')" onmouseout="rh()">Th13</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="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Maroon" title="c4">Z</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">Y</font></span><br/>

<a NAME="E6:76"/>
<font color="Maroon" title="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Maroon" title="c4">Z</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <font color="Maroon" title="c4">Z</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Maroon" title="c4">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="ref" href="pboole.html#T53" target="_self" onmouseover="rs('pboole/T53')" onmouseout="rh()">Th53</a></span>;<br/>
<a NAME="E7:76"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A3</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Maroon" title="c4">Z</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">Y</font> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Maroon" title="c4">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T75" target="_self" onmouseover="rs('pboole/T75')" onmouseout="rh()">Th75</a></span>;<br/>
<a NAME="E8:76"/>
<font color="Maroon" title="c3">Y</font> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Maroon" title="c4">Z</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T56" target="_self" onmouseover="rs('pboole/T56')" onmouseout="rh()">Th56</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:76"/>
<font color="Maroon" title="c2">X</font> <a href="pboole.html#K4" title="PBOOLE:func.4">(\)</a> <font color="Maroon" title="c4">Z</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:76"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="pboole.html#T13" target="_self" onmouseover="rs('pboole/T13')" onmouseout="rh()">Th13</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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