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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">FB</font> being   <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">B</font> holds <br/> ( <font color="Olive" title="b1">FB</font> is <a href="filter_0.html#V1" title="FILTER_0:attr.1">being_ultrafilter</a> iff ( <font color="Olive" title="b1">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> &amp; ( for <font color="Olive" title="b2">a</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="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">FB</font> or <font color="Olive" title="b2">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="Olive" title="b1">FB</font> ) ) ) )</span><br/><b>let </b><font color="Maroon" title="c2">FB</font> be    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> iff ( <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> &amp; ( for <font color="Olive" title="b1">a</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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> or <font color="Olive" title="b1">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/>



<b>thus </b><a NAME="E1:60"/>
( <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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">B</font> &amp; ( for <font color="Olive" title="b1">a</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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> or <font color="Olive" title="b1">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> ) ) ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> &amp; ( for <font color="Olive" title="b1">a</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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> or <font color="Olive" title="b1">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> ) ) implies <font color="Maroon" title="c2">FB</font> is <a href="filter_0.html#V1" title="FILTER_0:attr.1">being_ultrafilter</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:60_1"/><i><font color="Green" title="E45">A1</font></i>: 
( <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> &amp; ( for <font color="Olive" title="b1">HB</font> being   <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">B</font>  st <font color="Maroon" title="c2">FB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">HB</font> &amp; <font color="Olive" title="b1">HB</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> holds <br/><font color="Maroon" title="c2">FB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">HB</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D4" target="_self" title="FILTER_0:def.4">FILTER_0:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> &amp; ( for <font color="Olive" title="b1">a</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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> or <font color="Olive" title="b1">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/>

<b>hence </b><a NAME="E2:60_1"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> or <font color="Olive" title="b1">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/>

<b>reconsider </b><font color="Maroon" title="c3">I</font> = <font color="Maroon" title="c1">B</font> as   <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_Lattice</a> <b>by </b><i><a class="ref" href="filter_0.html#T40" target="_self" title="FILTER_0:th.40">Th40</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">FI</font> = <font color="Maroon" title="c2">FB</font> as    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c3">I</font> ;<br/>
<b>let </b><font color="Maroon" title="c5">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">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="c5">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/>

<b>assume </b><a NAME="E5:60_1"/><i><font color="Green" title="E46">A2</font></i>: 
not <font color="Maroon" title="c5">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">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/>

<b>reconsider </b><font color="Maroon" title="c6">a'</font> = <font color="Maroon" title="c5">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">I</font> ;<br/>
<a NAME="E7:60_1"/><i><font color="Green" title="E47">A3</font></i>: 
( <font color="Maroon" title="c2">FB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">FB</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> &amp; <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">FB</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> &amp; <font color="Maroon" title="c2">FB</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">FB</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> )
 
<b>by </b><i><a class="ref" href="filter_0.html#D5" target="_self" title="FILTER_0:def.5">Def5</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E8:60_1"/><b>then </b>
( <font color="Maroon" title="c2">FB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">FB</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">FB</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:60_1"/><b>then </b>
<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#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">FB</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
 
<b>by </b><i><a class="txt" href="#E5:60_1"><i><font color="Green" title="E46">A2</font></i></a></i>;<br/>
<a NAME="E10:60_1"/><b>then </b>
<span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">FB</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">B</font>
 
<b>by </b><i><a class="txt" href="#E1:60_1"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="#E7:60_1"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:60_1"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">FB</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T30" target="_self" title="FILTER_0:th.30">Th30</a></i>;<br/>
<a NAME="E12:60_1"/><b>then </b>
( <font color="Maroon" title="c6">a'</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">FI</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">a'</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <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> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T26" target="_self" title="FILTER_0:th.26">Th26</a>, <a class="ref" href="filter_0.html#T47" target="_self" title="FILTER_0:th.47">Th47</a></i>;<br/>
<a NAME="E13:60_1"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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> &amp; <font color="Maroon" title="c5">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T53" target="_self" title="FILTER_0:th.53">Th53</a>, <a class="ref" href="filter_0.html#T55" target="_self" title="FILTER_0:th.55">Th55</a></i>;<br/>
<b>hence </b><a NAME="E14:60_1"/>
<font color="Maroon" title="c5">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>
 <b>by </b><i><a class="ref" href="lattices.html#T17" title="LATTICES:th.17">LATTICES:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><b>that </b><br/><a NAME="E2:60"/><i><font color="Green" title="E45">A4</font></i>: 
<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>
 <b>and </b><br/><a NAME="E3:60"/><i><font color="Green" title="E46">A5</font></i>: 
for <font color="Olive" title="b1">a</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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> or <font color="Olive" title="b1">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><a NAME="E4:60"/>
<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>
 <b>by </b><i><a class="txt" href="#E2:60"><i><font color="Green" title="E45">A4</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D4" target="_self" title="FILTER_0:def.4">FILTER_0:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">H</font> being   <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">B</font>  st <font color="Maroon" title="c2">FB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b1">H</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> holds <br/><font color="Maroon" title="c2">FB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H</font></span><br/>

<b>let </b><font color="Maroon" title="c3">HB</font> be    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">FB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">HB</font> &amp; <font color="Maroon" title="c3">HB</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> implies <font color="Maroon" title="c2">FB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">HB</font> )</span><br/>

<b>assume </b><a NAME="E5:60"/><i><font color="Green" title="E47">A6</font></i>: 
( <font color="Maroon" title="c2">FB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">HB</font> &amp; <font color="Maroon" title="c3">HB</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> &amp; <font color="Maroon" title="c2">FB</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">HB</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:60"/><b>then </b>
not for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">HB</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:60"/><i><font color="Green" title="E48">A7</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">HB</font> &amp; not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> )
 <b>by </b><i><a class="txt" href="#E5:60"><i><font color="Green" title="E47">A6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">HB</font> <b>by </b><i><a class="txt" href="#E8:60"><i><font color="Green" title="E48">A7</font></i></a></i>;<br/>
<a NAME="E10:60"/>
<font color="Maroon" title="c5">x</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>
 
<b>by </b><i><a class="txt" href="#E3:60"><i><font color="Green" title="E46">A5</font></i></a>, <a class="txt" href="#E8:60"><i><font color="Green" title="E48">A7</font></i></a></i>;<br/>
<a NAME="E11:60"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">HB</font> &amp; <font color="Maroon" title="c1">B</font> is   <a href="lattices.html#NM4" title="LATTICES:NM.4">0_Lattice</a> )
 
<b>by </b><i><a class="txt" href="#E5:60"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="filter_0.html#D1" target="_self" title="FILTER_0:def.1">Def1</a></i>;<br/>
<a NAME="E12:60"/><b>then </b>
(  <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="c3">HB</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">B</font></span>)</span></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> &amp; <font color="Maroon" title="c1">B</font> is   <a href="lattices.html#NM4" title="LATTICES:NM.4">0_Lattice</a> &amp; <font color="Maroon" title="c3">HB</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="c3">HB</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T20" target="_self" title="FILTER_0:th.20">Th20</a>, <a class="ref" href="filter_0.html#T26" target="_self" title="FILTER_0:th.26">Th26</a>, <a class="ref" href="lattices.html#T47" title="LATTICES:th.47">LATTICES:47</a></i>;<br/>
<b>hence </b><a NAME="E13:60"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:60"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="filter_0.html#T31" target="_self" title="FILTER_0:th.31">Th31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
