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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>



<span class="kw">assume </span><a NAME="E1:43"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopclset.html#D8" target="_self" title="LOPCLSET:def.8">Def8</a></span>;<br/>
<a NAME="E3:43"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c3">p</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="lopclset.html#E38"><span class="lab"><font color="Green" title="E19">Lm1</font></span></a></span>;<br/>
<a NAME="E4:43"/>
<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="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><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="E1:43_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopclset.html#D8" target="_self" title="LOPCLSET:def.8">Def8</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:43_1"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="lopclset.html#T26" target="_self" title="LOPCLSET:th.26">Th26</a></span>;<br/>
<a NAME="E4:43_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="lopclset.html#T28" target="_self" title="LOPCLSET:th.28">Th28</a></span>;<br/>
<a NAME="E5:43_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopclset.html#T26" target="_self" title="LOPCLSET:th.26">Th26</a></span>;<br/>
<a NAME="E6:43_1"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="lopclset.html#E38"><span class="lab"><font color="Green" title="E19">Lm1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:43_1"/>
<font color="Maroon" title="c3">p</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:43"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E28">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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