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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>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font>;<br/><b>assume </b><a NAME="E1:31_1"/><i><font color="Green" title="E23">A1</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="c2">a</font> <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> <font color="Maroon" title="c3">b</font>
 ;<br/><b>assume </b><a NAME="E2:31_1"/>
not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">UF</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="E4:31_1"/><i><font color="Green" title="E24">A2</font></i>: 
( <font color="Maroon" title="c4">UF</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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">UF</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">UF</font> ) or ( not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">UF</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">UF</font> ) ) )
 <b>by </b><i><a class="ref" href="filter_0.html#T60" title="FILTER_0:th.60">FILTER_0:60</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_1_1"/>
( ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">UF</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">UF</font> ) or ( not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">UF</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">UF</font> ) )
 </span><b>by </b><i><a class="txt" href="lopclset.html#E4:31_1"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1"><b>case </b></a><a NAME="E1:31_1_1_1_1"/><i><font color="Green" title="E25">A3</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">UF</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">UF</font> )
 ;<br/><div class="add"><a NAME="E2:31_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">UF</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#E4:31_1"><i><font color="Green" title="E24">A2</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="E3:31_1_1_1_1"/>
<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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="c3">b</font>
 <b>by </b><i><a class="txt" href="lopclset.html#E1:31_1_1_1_1"><i><font color="Green" title="E25">A3</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2"><b>case </b></a><a NAME="E1:31_1_1_1_2"/>
( not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">UF</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">UF</font> )
 ;<br/><div class="add"><a NAME="E2:31_1_1_1_2"/><b>then </b>
not <font color="Maroon" title="c4">UF</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="lopclset.html#T21" target="_self" title="LOPCLSET:th.21">Th21</a></i>;<br/><b>hence </b><a NAME="E3:31_1_1_1_2"/>
<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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="c3">b</font>
 <b>by </b><i><a class="txt" href="lopclset.html#E4:31_1"><i><font color="Green" title="E24">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>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:31_1"/>
contradiction
 <b>by </b><i><a class="txt" href="lopclset.html#E1:31_1"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:31"/>
 <a href="lopclset.html#K9" title="LOPCLSET:func.9">UFilter</a> <font color="Maroon" title="c1">BL</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="group_6.html#T1" title="GROUP_6:th.1">GROUP_6:1</a></i>;<br/>


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