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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">M</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being  <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of <font color="Olive" title="b1">M</font>  st <font color="Olive" title="b2">g</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a> &amp; <font color="Olive" title="b3">h</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a> &amp; <font color="Olive" title="b2">g</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Olive" title="b3">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">h</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Olive" title="b2">g</font> holds <br/><font color="Olive" title="b2">g</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Olive" title="b3">h</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a></span><br/><b>let </b><font color="Maroon" title="c2">M</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"> for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> being  <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b1">g</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a> &amp; <font color="Olive" title="b2">h</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a> &amp; <font color="Olive" title="b1">g</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Olive" title="b2">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">h</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Olive" title="b1">g</font> holds <br/><font color="Olive" title="b1">g</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Olive" title="b2">h</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a></span><br/><b>let </b><font color="Maroon" title="c3">g</font>, <font color="Maroon" title="c4">h</font> be   <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of <font color="Maroon" title="c2">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">g</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a> &amp; <font color="Maroon" title="c4">h</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a> &amp; <font color="Maroon" title="c3">g</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Maroon" title="c4">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">h</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Maroon" title="c3">g</font> implies <font color="Maroon" title="c3">g</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Maroon" title="c4">h</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:65"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c3">g</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a>
 <b>and </b><br/><a NAME="E2:65"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c4">h</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a>
 <b>and </b><br/><a NAME="E3:65"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Maroon" title="c4">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">h</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Maroon" title="c3">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">g</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Maroon" title="c4">h</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a></span><br/>

<b>let </b><font color="Maroon" title="c5">X</font> be    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D14" target="_self" title="CLOSURE2:def.14">CLOSURE2:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">X</font></span>)</span></span><br/>

<a NAME="E4:65"/><i><font color="Green" title="E32">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:65"/><i><font color="Green" title="E33">A5</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">X</font> = 
<font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">X</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:65"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:65"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="closure2.html#D14" target="_self" title="CLOSURE2:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">X</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:65"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="closure2.html#D14" target="_self" title="CLOSURE2:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:65"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="#E4:65"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">X</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E5:65"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:65"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="closure2.html#K8" title="CLOSURE2:func.8">*</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">X</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:65"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</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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