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<b>let </b><font color="Maroon" title="c1">BL</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="lattices.html#NM8" title="LATTICES:NM.8">B_Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <span class="p1">(<span class="default"><a href="lopclset.html#K11" title="LOPCLSET:func.11">StoneSpace</a> <font color="Maroon" title="c1">BL</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <span class="p1">(<span class="default"><a href="lopclset.html#K11" title="LOPCLSET:func.11">StoneSpace</a> <font color="Maroon" title="c1">BL</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <span class="p1">(<span class="default"><a href="lopclset.html#K11" title="LOPCLSET:func.11">StoneSpace</a> <font color="Maroon" title="c1">BL</font></span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c3">p</font> = <font color="Maroon" title="c2">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K11" title="LOPCLSET:func.11">StoneSpace</a> <font color="Maroon" title="c1">BL</font></span>)</span> <b>by </b><i><a class="ref" href="lopclset.html#D8" target="_self" title="LOPCLSET:def.8">Def8</a></i>;<br/>
<a NAME="E3:40"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c3">p</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="lopclset.html#E1:40"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="lopclset.html#E38"><i><font color="Green" title="E21">Lm1</font></i></a></i>;<br/>
<a NAME="E4:40"/>
<font color="Maroon" title="c3">p</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c4">ST</font> =  <a href="lopclset.html#K11" title="LOPCLSET:func.11">StoneSpace</a> <font color="Maroon" title="c1">BL</font>;<br/>
<a NAME="E1:40_1"/><i><font color="Green" title="E31">A3</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="lopclset.html#K11" title="LOPCLSET:func.11">StoneSpace</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c1">BL</font>
 
<b>by </b><i><a class="ref" href="lopclset.html#D8" target="_self" title="LOPCLSET:def.8">Def8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font><b> such that </b><br/><a NAME="E3:40_1"/><i><font color="Green" title="E32">A4</font></i>: 
<span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="lopclset.html#E1:40"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="lopclset.html#T26" target="_self" title="LOPCLSET:th.26">Th26</a></i>;<br/>
<a NAME="E4:40_1"/>
<font color="Maroon" title="c3">p</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="lopclset.html#E1:40_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="lopclset.html#E3:40_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="lopclset.html#T28" target="_self" title="LOPCLSET:th.28">Th28</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font><b> such that </b><br/><a NAME="E6:40_1"/><i><font color="Green" title="E33">A5</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  &amp; <span class="p1">(<span class="default"><a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 ;<br/>
<a NAME="E7:40_1"/>
<font color="Maroon" title="c3">p</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopclset.html#K10" title="LOPCLSET:func.10">StoneR</a> <font color="Maroon" title="c1">BL</font>
 
<b>by </b><i><a class="txt" href="lopclset.html#E6:40_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="lopclset.html#T26" target="_self" title="LOPCLSET:th.26">Th26</a></i>;<br/>
<a NAME="E8:40_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="lopclset.html#E38"><i><font color="Green" title="E21">Lm1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:40_1"/>
<font color="Maroon" title="c3">p</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:40"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopclset.html#K1" title="LOPCLSET:func.1">OpenClosedSet</a> <span class="p1">(<span class="default"><a href="lopclset.html#K11" title="LOPCLSET:func.11">StoneSpace</a> <font color="Maroon" title="c1">BL</font></span>)</span>
 <b>by </b><i><a class="txt" href="lopclset.html#E3:40"><i><font color="Green" title="E30">A2</font></i></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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