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<b>let </b><font color="Maroon" title="c1">BL</font> be  non <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a> <a href="lattices.html#NM8" title="LATTICES:NM.8">B_Lattice</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font>;<br/>



<div><b>hereby </b> <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="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<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="c2">a</font></span>)</span>
 ;<br/><a NAME="E2:29_1"/><b>then </b><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c1">BL</font> &amp; not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c4">F</font> being    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">BL</font><b> such that </b><br/><a NAME="E4:29_1"/><i><font color="Green" title="E28">A2</font></i>: 
( <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c4">F</font> <a href="filter_0.html#NR1" title="FILTER_0:NR.1">is_ultrafilter</a>  )
 ;<br/><a NAME="E5:29_1"/>
not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="lopclset.html#E2:29_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="lopclset.html#E4:29_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lopclset.html#T21" target="_self" title="LOPCLSET:th.21">Th21</a></i>;<br/><a NAME="E6:29_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="lopclset.html#E4:29_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="filter_0.html#T59" title="FILTER_0:th.59">FILTER_0:59</a></i>;<br/><b>hence </b><a NAME="E7:29_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span>
 <b>by </b><i><a class="txt" href="lopclset.html#E4:29_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lopclset.html#T21" target="_self" title="LOPCLSET:th.21">Th21</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>
<div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">F</font> being    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">BL</font><b> such that </b><br/><a NAME="E3:29_2"/><i><font color="Green" title="E27">A3</font></i>: 
( <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c4">F</font> <a href="filter_0.html#NR1" title="FILTER_0:NR.1">is_ultrafilter</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> )
 <b>by </b><i><a class="ref" href="lopclset.html#T20" target="_self" title="LOPCLSET:th.20">Th20</a></i>;<br/><a NAME="E4:29_2"/><i><font color="Green" title="E28">A4</font></i>: 
not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="lopclset.html#E3:29_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="filter_0.html#T59" title="FILTER_0:th.59">FILTER_0:59</a></i>;<br/><a NAME="E5:29_2"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="txt" href="lopclset.html#E3:29_2"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/><a NAME="E6:29_2"/>
not <font color="Maroon" title="c4">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="lopclset.html#E4:29_2"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="lopclset.html#T21" target="_self" title="LOPCLSET:th.21">Th21</a></i>;<br/><b>hence </b><a NAME="E7:29_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K7" title="LOPCLSET:func.7">ultraset</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<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="c2">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="lopclset.html#E3:29_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="lopclset.html#E5:29_2"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E3:29"/>
verum
 ;<br/>


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