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<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> holds <br/> ex <font color="Olive" title="b3">IB</font> being   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">B</font> st <br/>( <font color="Olive" title="b3">IB</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">max-ideal</a>  &amp; ( ( <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">IB</font> &amp;  not <font color="Olive" title="b2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">IB</font> ) or (  not <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">IB</font> &amp; <font color="Olive" title="b2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">IB</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> implies  ex <font color="Olive" title="b1">IB</font> being   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">B</font> st <br/>( <font color="Olive" title="b1">IB</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">max-ideal</a>  &amp; ( ( <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">IB</font> &amp;  not <font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">IB</font> ) or (  not <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">IB</font> &amp; <font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">IB</font> ) ) ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:78"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">IB</font> being   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">B</font> st <br/>( <font color="Olive" title="b1">IB</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">max-ideal</a>  &amp; ( ( <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">IB</font> &amp;  not <font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">IB</font> ) or (  not <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">IB</font> &amp; <font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">IB</font> ) ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">FB</font> being   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">B</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:78"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c4">FB</font> is  <a href="filter_0.html#V1" title="FILTER_0:attr.1">being_ultrafilter</a> 
 <span class="kw">and </span><br/><a NAME="E4:78"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
( ( <font color="Maroon" title="c2">a</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">FB</font> &amp;  not <font color="Maroon" title="c3">b</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">FB</font> ) or (  not <font color="Maroon" title="c2">a</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">FB</font> &amp; <font color="Maroon" title="c3">b</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">FB</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T47" onmouseover="rs('filter_0/T47')" onmouseout="rh()">FILTER_0:47</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c5">IB</font> =  <a href="filter_2.html#K4" title="FILTER_2:func.4">.:</a> <font color="Maroon" title="c4">FB</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">IB</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">max-ideal</a>  &amp; ( ( <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">IB</font> &amp;  not <font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">IB</font> ) or (  not <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">IB</font> &amp; <font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">IB</font> ) ) )</span><br/>

<a NAME="E5:78"/>
<font color="Maroon" title="c5">IB</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c4">FB</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E6:78"/>
<font color="Maroon" title="c5">IB</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">max-ideal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="filter_2.html#T32" target="_self" onmouseover="rs('filter_2/T32')" onmouseout="rh()">Th32</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">IB</font> &amp;  not <font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">IB</font> ) or (  not <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">IB</font> &amp; <font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">IB</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E7:78"/>
( ( <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">IB</font> &amp;  not <font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">IB</font> ) or (  not <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">IB</font> &amp; <font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">IB</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78"><span class="lab"><font color="Green" title="E30">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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