<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">B</font> be   <a href="lattices.html#NM8" title="LATTICES:NM.8">B_Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">FB</font> being   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">B</font> holds <br/> ( ( <font color="Olive" title="b1">FB</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="filter_0.html#K1" title="FILTER_0:func.1">&lt;.</a><span class="default"><font color="Maroon" title="c1">B</font></span><a href="filter_0.html#K1" title="FILTER_0:func.1">.)</a></span> &amp; <font color="Olive" title="b1">FB</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  ) iff <font color="Olive" title="b1">FB</font> is  <a href="filter_0.html#V1" title="FILTER_0:attr.1">being_ultrafilter</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">FB</font> be   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">FB</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="filter_0.html#K1" title="FILTER_0:func.1">&lt;.</a><span class="default"><font color="Maroon" title="c1">B</font></span><a href="filter_0.html#K1" title="FILTER_0:func.1">.)</a></span> &amp; <font color="Maroon" title="c2">FB</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  ) iff <font color="Maroon" title="c2">FB</font> is  <a href="filter_0.html#V1" title="FILTER_0:attr.1">being_ultrafilter</a>  )</span><br/>



<span class="kw">thus </span><a NAME="E1:60"/>
( <font color="Maroon" title="c2">FB</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="filter_0.html#K1" title="FILTER_0:func.1">&lt;.</a><span class="default"><font color="Maroon" title="c1">B</font></span><a href="filter_0.html#K1" title="FILTER_0:func.1">.)</a></span> &amp; <font color="Maroon" title="c2">FB</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  implies <font color="Maroon" title="c2">FB</font> is  <a href="filter_0.html#V1" title="FILTER_0:attr.1">being_ultrafilter</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">FB</font> is  <a href="filter_0.html#V1" title="FILTER_0:attr.1">being_ultrafilter</a>  implies ( <font color="Maroon" title="c2">FB</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="filter_0.html#K1" title="FILTER_0:func.1">&lt;.</a><span class="default"><font color="Maroon" title="c1">B</font></span><a href="filter_0.html#K1" title="FILTER_0:func.1">.)</a></span> &amp; <font color="Maroon" title="c2">FB</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:60_1"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
<font color="Maroon" title="c2">FB</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="filter_0.html#K1" title="FILTER_0:func.1">&lt;.</a><span class="default"><font color="Maroon" title="c1">B</font></span><a href="filter_0.html#K1" title="FILTER_0:func.1">.)</a></span>
 <span class="kw">and </span><br/><a NAME="E2:60_1"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font> holds <br/> ( <font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> iff ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> or <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> ) )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="filter_0.html#D5" target="_self" title="FILTER_0:def.5">FILTER_0:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">FB</font> is  <a href="filter_0.html#V1" title="FILTER_0:attr.1">being_ultrafilter</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> implies <font color="Maroon" title="c3">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> )</span><br/><span class="kw">assume </span><a NAME="E1:60_1_1"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
 not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font></span><br/><a NAME="E2:60_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></span>;<br/><a NAME="E3:60_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T11" target="_self" title="FILTER_0:th.11">Th12</a></span>;<br/><span class="kw">hence </span><a NAME="E4:60_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E1:60_1_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:60_1"/>
<font color="Maroon" title="c2">FB</font> is  <a href="filter_0.html#V1" title="FILTER_0:attr.1">being_ultrafilter</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="filter_0.html#T44" target="_self" title="FILTER_0:th.44">Th57</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:60"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
<font color="Maroon" title="c2">FB</font> is  <a href="filter_0.html#V1" title="FILTER_0:attr.1">being_ultrafilter</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">FB</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="filter_0.html#K1" title="FILTER_0:func.1">&lt;.</a><span class="default"><font color="Maroon" title="c1">B</font></span><a href="filter_0.html#K1" title="FILTER_0:func.1">.)</a></span> &amp; <font color="Maroon" title="c2">FB</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  )</span><br/>

<span class="kw">hence </span><a NAME="E3:60"/>
<font color="Maroon" title="c2">FB</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="filter_0.html#K1" title="FILTER_0:func.1">&lt;.</a><span class="default"><font color="Maroon" title="c1">B</font></span><a href="filter_0.html#K1" title="FILTER_0:func.1">.)</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T44" target="_self" title="FILTER_0:th.44">Th57</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">FB</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="filter_0.html#D5" target="_self" title="FILTER_0:def.5">FILTER_0:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font> holds <br/> ( <font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> iff ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> or <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> iff ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> or <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> ) )</span><br/>



<a NAME="E4:60"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
<font color="Maroon" title="c2">FB</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="filter_0.html#T44" target="_self" title="FILTER_0:th.44">Th57</a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:60"/>
(  not <font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> or <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> or <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> or <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> ) implies <font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:60_2"/><span class="lab"><font color="Green" title="E43">A6</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font>
 <span class="kw">and </span><br/><a NAME="E2:60_2"/><span class="lab"><font color="Green" title="E44">A7</font></span>: 
(  not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> &amp;  not <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:60_2"/>
( <font color="Maroon" title="c3">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> &amp; <font color="Maroon" title="c4">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E2:60_2"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="ref" href="filter_0.html#T44" target="_self" title="FILTER_0:th.44">Th57</a></span>;<br/>
<a NAME="E4:60_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T8" target="_self" title="FILTER_0:th.8">Th8</a></span>;<br/>
<a NAME="E5:60_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T24" title="LATTICES:th.24">LATTICES:24</a></span>;<br/>
<a NAME="E6:60_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_2"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="ref" href="filter_0.html#T8" target="_self" title="FILTER_0:th.8">Th8</a></span>;<br/>
<a NAME="E7:60_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A8</font></span>: 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T20" title="LATTICES:th.20">LATTICES:20</a></span>;<br/>
<a NAME="E8:60_2"/>
<font color="Maroon" title="c2">FB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c2">FB</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T21" target="_self" title="FILTER_0:th.21">Th26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:60_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E7:60_2"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="ref" href="filter_0.html#T25" target="_self" title="FILTER_0:th.25">Th31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E6:60"/>
( ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> or <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> ) implies <font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T10" target="_self" title="FILTER_0:th.10">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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