<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.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#M1" title="PBOOLE:mode.1">Element</a> of  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">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#K2" title="CLOSURE1:func.2">..</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#K2" title="CLOSURE1:func.2">..</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="pralg_1.html#K16" title="PRALG_1:func.16">..</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/><span class="kw">let </span><font color="Maroon" title="c2">M</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#M1" title="PBOOLE:mode.1">Element</a> of  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">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#K2" title="CLOSURE1:func.2">..</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#K2" title="CLOSURE1:func.2">..</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="pralg_1.html#K16" title="PRALG_1:func.16">..</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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E</font>, <font color="Olive" title="b2">T</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">Element</a> of  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">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#K2" title="CLOSURE1:func.2">..</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#K2" title="CLOSURE1:func.2">..</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="pralg_1.html#K16" title="PRALG_1:func.16">..</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/><span class="kw">let </span><font color="Maroon" title="c4">E</font>, <font color="Maroon" title="c5">T</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">Element</a> of  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a> <font color="Maroon" title="c2">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K2" title="CLOSURE1:func.2">..</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#K2" title="CLOSURE1:func.2">..</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="pralg_1.html#K16" title="PRALG_1:func.16">..</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#K5" title="MSSUBFAM:func.5">bool</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mssubfam.html#T12" title="MSSUBFAM:th.12">MSSUBFAM:12</a></span>;<br/>
<a NAME="E2:29"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mboolean.html#T1" title="MBOOLEAN:th.1">MBOOLEAN:1</a></span>;<br/>
<a NAME="E3:29"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mboolean.html#T15" title="MBOOLEAN:th.15">MBOOLEAN:15</a></span>;<br/>
<a NAME="E4:29"/><span class="kw">then </span>
<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#K5" title="MSSUBFAM:func.5">bool</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mboolean.html#T1" title="MBOOLEAN:th.1">MBOOLEAN:1</a></span>;<br/>
<a NAME="E5:29"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<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#M1" title="PBOOLE:mode.1">Element</a> of  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mssubfam.html#T11" title="MSSUBFAM:th.11">MSSUBFAM:11</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:29"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</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#K2" title="CLOSURE1:func.2">..</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="pralg_1.html#K16" title="PRALG_1:func.16">..</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/>

<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</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#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c5">T</font></span>)</span> = 
<font color="Maroon" title="c3">P</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="closure1.html#D4" target="_self" title="CLOSURE1:def.4">Def5</a></span>
<br/>.= 
<font color="Maroon" title="c3">P</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T67" title="PBOOLE:th.67">PBOOLE:67</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</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#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c5">T</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E6:29"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="closure1.html#D4" target="_self" title="CLOSURE1:def.4">Def5</a></span>
;<br/>
<a NAME="E8:29"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</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="pralg_1.html#K16" title="PRALG_1:func.16">..</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#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c5">T</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T14" title="PBOOLE:th.14">PBOOLE:14</a></span>;<br/>
<a NAME="E9:29"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</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#K2" title="CLOSURE1:func.2">..</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="pralg_1.html#K16" title="PRALG_1:func.16">..</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#K2" title="CLOSURE1:func.2">..</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#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c5">T</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T53" title="PBOOLE:th.53">PBOOLE:53</a></span>;<br/>
<a NAME="E10:29"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</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#K2" title="CLOSURE1:func.2">..</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="pralg_1.html#K16" title="PRALG_1:func.16">..</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#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c5">T</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T75" title="PBOOLE:th.75">PBOOLE:75</a></span>;<br/>
<a NAME="E11:29"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</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#K2" title="CLOSURE1:func.2">..</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="pralg_1.html#K16" title="PRALG_1:func.16">..</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 class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T56" title="PBOOLE:th.56">PBOOLE:56</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:29"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</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#K2" title="CLOSURE1:func.2">..</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="pralg_1.html#K16" title="PRALG_1:func.16">..</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E10:29"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="pboole.html#T13" title="PBOOLE:th.13">PBOOLE:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
