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<b>let </b><font color="Maroon">B</font> be   <a href="lattices.html#NM8">B_Lattice</a>;<br/><b>let </b><font color="Maroon">FB</font> be    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">B</font>;<br/>



<b>thus </b><a NAME="E1:62"/>
( <font color="Maroon">FB</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="filter_0.html#K1" target="_self">&lt;.</a><span class="default"><font color="Maroon">B</font></span><a href="filter_0.html#K1" target="_self">.)</a></span> &amp; <font color="Maroon">FB</font> is <a href="filter_0.html#V2" target="_self">prime</a> implies <font color="Maroon">FB</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:62_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">FB</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="filter_0.html#K1" target="_self">&lt;.</a><span class="default"><font color="Maroon">B</font></span><a href="filter_0.html#K1" target="_self">.)</a></span>
 <b>and </b><br/><a NAME="E2:62_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> holds <br/> ( <font color="Olive">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> iff ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> or <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D6" target="_self">FILTER_0:def 6</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>;<br/><b>assume </b><a NAME="E1:62_1_1"/><i><font color="Green">E43</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font>
 ;<br/><a NAME="E2:62_1_1"/>
( <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> is   <a href="lattices.html#NM5">1_Lattice</a> )
 <b>by </b><i><a class="ref" href="lattices.html#T48">LATTICES:48</a></i>;<br/><a NAME="E3:62_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:62_1_1"/>
<font color="Maroon">a</font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font>
 <b>by </b><i>, <a class="ref" href="filter_0.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:62_1"/>
<font color="Maroon">FB</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a> 
 <b>by </b><i><a class="txt" href="filter_0.html#E2:62"><i><font color="Green">E44</font></i></a>, </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:62"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">FB</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a> 
 ;<br/>

<a NAME="E3:62"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">FB</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp; ( for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>  holds <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> or <font color="Olive">a</font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> ) ) )
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:62"/>
<font color="Maroon">FB</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="filter_0.html#K1" target="_self">&lt;.</a><span class="default"><font color="Maroon">B</font></span><a href="filter_0.html#K1" target="_self">.)</a></span>
 <b>by </b><i><a class="txt" href="filter_0.html#E2:62"><i><font color="Green">E44</font></i></a>, </i>;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D6" target="_self">FILTER_0:def 6</a><br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>;<br/>



<b>thus </b><a NAME="E5:62"/>
(  not <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> or <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:62_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> )
 ;<br/>

<a NAME="E2:62_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> &amp; <font color="Maroon">b</font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> )
 
<b>by </b><i><a class="txt" href="filter_0.html#E2:62"><i><font color="Green">E44</font></i></a>, </i>;<br/>
<a NAME="E3:62_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:62_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T51">LATTICES:51</a></i>;<br/>
<a NAME="E5:62_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K7">`</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font>
 
<b>by </b><i><a class="ref" href="filter_0.html#T10" target="_self">Th10</a>, </i>;<br/>
<a NAME="E6:62_2"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T47">LATTICES:47</a></i>;<br/>
<a NAME="E7:62_2"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_0.html#K1" target="_self">&lt;.</a><span class="default"><font color="Maroon">B</font></span><a href="filter_0.html#K1" target="_self">.)</a></span> &amp; <span class="p1"><a href="filter_0.html#K1" target="_self">&lt;.</a><span class="default"><font color="Maroon">B</font></span><a href="filter_0.html#K1" target="_self">.)</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_0.html#K2" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">B</font></span>)</span></span><a href="filter_0.html#K2" target="_self">.)</a></span> &amp; <font color="Maroon">B</font> is   <a href="lattices.html#NM4">0_Lattice</a> &amp; <font color="Maroon">FB</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">FB</font></span><a href="filter_0.html#K3" target="_self">.)</a></span> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:62_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="filter_0.html#T11" target="_self">Th11</a>, </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:62"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> or <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> ) implies <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> )
 <b>by </b><i/>;<br/>


</div>
