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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="realset2.html#V3">trivial</a> <a href="lattices.html#NM8">B_Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<a NAME="E1:18"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K8" target="_self">UFilter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K8" target="_self">UFilter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:18_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">c<sub>2</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="lopclset.html#D6" target="_self">Def6</a></i>;<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>2</sub></font><b> such that </b><br/><a NAME="E4:18_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <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>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E5:18_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <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>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="lopclset.html#E4:18_1"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:18"/>
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K8" target="_self">UFilter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:18_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">c<sub>2</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E3:18_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K8" target="_self">UFilter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="lopclset.html#D6" target="_self">Def6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K8" target="_self">UFilter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> iff ex <font color="Olive">b<sub>1</sub></font> being   <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="lopclset.html#E1:18"><i><font color="Green">E20</font></i></a></i>;<br/>


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