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<b>let </b><font color="Maroon">L</font> be  non <a href="realset2.html#V3">trivial</a> <a href="waybel_1.html#V11">Boolean</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">F</font> be  <a href="tex_2.html#V1">proper</a> <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">L</font>;<br/>



<a NAME="E1:40"/>
 <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T23">WAYBEL_4:23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">I</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:40_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:40_1"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E4:40"/><b>then </b>
<font color="Maroon">I</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">G</font> being   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:40"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">G</font> is <a href="waybel_7.html#V2" target="_self">prime</a> &amp; <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">I</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_2"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/><a NAME="E2:40_2"/><b>then </b>
not  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E3:40_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:40"/><b>then </b>
<font color="Maroon">G</font> is <a href="tex_2.html#V1">proper</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:40"/>
( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> is <a href="waybel_7.html#V3" target="_self">ultra</a> )
 <b>by </b><i>, </i>;<br/>


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