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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">P</font> being  <a href="closure1.html#NM1" title="CLOSURE1:NM.1">MSSetOp</a> of <font color="Olive" title="b1">M</font><br/> for <font color="Olive" title="b3">E</font>, <font color="Olive" title="b4">T</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Olive" title="b1">M</font>  st <font color="Olive" title="b2">P</font> is <a href="closure1.html#V4" title="CLOSURE1:attr.4">topological</a> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b3">E</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b4">T</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b2">P</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Olive" title="b4">T</font></span>)</span></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">P</font> being  <a href="closure1.html#NM1" title="CLOSURE1:NM.1">MSSetOp</a> of <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b2">E</font>, <font color="Olive" title="b3">T</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b1">P</font> is <a href="closure1.html#V4" title="CLOSURE1:attr.4">topological</a> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</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="b1">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</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="b1">P</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Olive" title="b3">T</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">P</font> be   <a href="closure1.html#NM1" title="CLOSURE1:NM.1">MSSetOp</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">E</font>, <font color="Olive" title="b2">T</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font>  st <font color="Maroon" title="c3">P</font> is <a href="closure1.html#V4" title="CLOSURE1:attr.4">topological</a> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</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="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b2">T</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">P</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Olive" title="b2">T</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">E</font>, <font color="Maroon" title="c5">T</font> be    <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">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="c3">P</font> is <a href="closure1.html#V4" title="CLOSURE1:attr.4">topological</a> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c4">E</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">T</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">P</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c5">T</font></span>)</span> )</span><br/>









<a NAME="E1:29"/>
<font color="Maroon" title="c4">E</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T12" title="MSSUBFAM:th.12">MSSUBFAM:12</a></i>;<br/>
<a NAME="E2:29"/><b>then </b>
<font color="Maroon" title="c4">E</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="mboolean.html#T1" title="MBOOLEAN:th.1">MBOOLEAN:1</a></i>;<br/>
<a NAME="E3:29"/><b>then </b>
<font color="Maroon" title="c4">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c5">T</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="mboolean.html#T16" title="MBOOLEAN:th.16">MBOOLEAN:16</a></i>;<br/>
<a NAME="E4:29"/><b>then </b>
<font color="Maroon" title="c4">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c5">T</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="mboolean.html#T1" title="MBOOLEAN:th.1">MBOOLEAN:1</a></i>;<br/>
<a NAME="E5:29"/><b>then </b><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c4">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c5">T</font> is    <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T11" title="MSSUBFAM:th.11">MSSUBFAM:11</a></i>;<br/>
<b>assume </b><a NAME="E6:29"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c3">P</font> is <a href="closure1.html#V4" title="CLOSURE1:attr.4">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="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c4">E</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">T</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">P</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c5">T</font></span>)</span></span><br/>

<b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c4">E</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">T</font></span>)</span> = 
<font color="Maroon" title="c3">P</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">T</font></span>)</span>
<b>by </b><i><a class="ref" href="closure1.html#D5" target="_self" title="CLOSURE1:def.5">Def5</a></i>
<br/>.= 
<font color="Maroon" title="c3">P</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c5">T</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">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="c3">P</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c5">T</font></span>)</span></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">T</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:29"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="#E6:29"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="closure1.html#D5" target="_self" title="CLOSURE1:def.5">Def5</a></i>
;<br/>
<a NAME="E8:29"/><b>then </b>
<font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c4">E</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c5">T</font></span>)</span></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">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="E9:29"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c4">E</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">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="c3">P</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c5">T</font></span>)</span></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">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="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">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="E10:29"/><b>then </b><i><font color="Green" title="E14">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c4">E</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">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="c3">P</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c5">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="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">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="E11:29"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c5">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="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">T</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">P</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c5">T</font></span>)</span>
 
<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="E12:29"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c4">E</font></span>)</span> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">T</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">P</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">E</font> <a href="pboole.html#K4" title="PBOOLE:func.4">\</a> <font color="Maroon" title="c5">T</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:29"><i><font color="Green" title="E14">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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