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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">E</font>, <font color="Olive" title="b3">T</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Olive" title="b1">M</font><br/> for <font color="Olive" title="b4">g</font> being  <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of <font color="Olive" title="b1">M</font><br/> for <font color="Olive" title="b5">A</font> being   <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="Olive" title="b1">M</font>  st <font color="Olive" title="b5">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Olive" title="b3">T</font> &amp; <font color="Olive" title="b4">g</font> is <a href="closure2.html#V10" title="CLOSURE2:attr.10">topological</a> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b2">E</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b3">T</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b4">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b5">A</font></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">E</font>, <font color="Olive" title="b2">T</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b3">g</font> being  <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b4">A</font> being   <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>  st <font color="Olive" title="b4">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Olive" title="b2">T</font> &amp; <font color="Olive" title="b3">g</font> is <a href="closure2.html#V10" title="CLOSURE2:attr.10">topological</a> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">E</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b2">T</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b4">A</font></span><br/><b>let </b><font color="Maroon" title="c3">E</font>, <font color="Maroon" title="c4">T</font> be    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <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"> for <font color="Olive" title="b1">g</font> being  <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b2">A</font> being   <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>  st <font color="Olive" title="b2">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c4">T</font> &amp; <font color="Olive" title="b1">g</font> is <a href="closure2.html#V10" title="CLOSURE2:attr.10">topological</a> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b2">A</font></span><br/><b>let </b><font color="Maroon" title="c5">g</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"> for <font color="Olive" title="b1">A</font> being   <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>  st <font color="Olive" title="b1">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c4">T</font> &amp; <font color="Maroon" title="c5">g</font> is <a href="closure2.html#V10" title="CLOSURE2:attr.10">topological</a> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">A</font></span><br/><b>let </b><font color="Maroon" title="c6">A</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c4">T</font> &amp; <font color="Maroon" title="c5">g</font> is <a href="closure2.html#V10" title="CLOSURE2:attr.10">topological</a> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c6">A</font> )</span><br/>











<b>assume </b><a NAME="E1:62"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c4">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">g</font> is <a href="closure2.html#V10" title="CLOSURE2:attr.10">topological</a> or <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c6">A</font> )</span><br/>

<b>assume </b><a NAME="E2:62"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c5">g</font> is <a href="closure2.html#V10" title="CLOSURE2:attr.10">topological</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="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c6">A</font></span><br/>

<b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span> = 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">E</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c4">T</font></span>)</span>
<b>by </b><i><a class="ref" href="closure2.html#D15" target="_self" title="CLOSURE2:def.15">Def15</a></i>
<br/>.= 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c4">T</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#T73" title="PBOOLE:th.73">PBOOLE:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span>
<b>by </b><i><a class="txt" href="closure2.html#E1:62"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="closure2.html#E2:62"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="closure2.html#D15" target="_self" title="CLOSURE2:def.15">Def15</a></i>
;<br/>
<a NAME="E4:62"/><b>then </b>
<font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">E</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#T16" title="PBOOLE:th.16">PBOOLE:16</a></i>;<br/>
<a NAME="E5:62"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#T59" title="PBOOLE:th.59">PBOOLE:59</a></i>;<br/>
<a NAME="E6:62"/><b>then </b><i><font color="Green" title="E31">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#T81" title="PBOOLE:th.81">PBOOLE:81</a></i>;<br/>
<a NAME="E7:62"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c6">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T62" title="PBOOLE:th.62">PBOOLE:62</a></i>;<br/>
<b>hence </b><a NAME="E8:62"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c5">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c6">A</font>
 <b>by </b><i><a class="txt" href="closure2.html#E6:62"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</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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