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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="realset2.html#V3">trivial</a> <a href="lattices.html#NM8">B_Lattice</a>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:32_1"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>assume </b><a NAME="E2:32_1"/>
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:32_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; ( ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> ) or ( not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> ) ) )
 <b>by </b><i><a class="ref" href="filter_0.html#T60">FILTER_0:60</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_1_1"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> ) or ( not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="lopclset.html#E4:32_1"><i><font color="Green">E30</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1"><b>case </b></a><a NAME="E1:32_1_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><div class="add"><a NAME="E2:32_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="lopclset.html#E4:32_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="lopclset.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E3:32_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="lopclset.html#E1:32_1_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="lopclset.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2"><b>case </b></a><a NAME="E1:32_1_1_1_2"/>
( not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><div class="add"><a NAME="E2:32_1_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="lopclset.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E3:32_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="lopclset.html#E4:32_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="lopclset.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:32_1"/>
contradiction
 <b>by </b><i><a class="txt" href="lopclset.html#E1:32_1"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:32"/>
 <a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">c<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="group_6.html#T1">GROUP_6:1</a></i>;<br/>


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