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<div class="add">

<b>let </b><font color="Maroon">L</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>thus </b><a NAME="E1:24"/>
( <font color="Maroon">x</font> is    <a href="filter_2.html#M2" target="_self">Ideal</a> of <font color="Maroon">L</font> implies <font color="Maroon">x</font> is    <a href="filter_2.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1"/>
<font color="Maroon">x</font> is    <a href="filter_2.html#M2" target="_self">Ideal</a> of <font color="Maroon">L</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">I</font> = <font color="Maroon">x</font> as    <a href="filter_2.html#M2" target="_self">Ideal</a> of <font color="Maroon">L</font> ;<br/>
<b>reconsider </b><font color="Maroon">I</font> = <font color="Maroon">I</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span> ;<br/>
<a NAME="E4:24_1"/>
<font color="Maroon">I</font> is    <a href="filter_2.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p'</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D1">FILTER_0:def 1</a><br/><b>let </b><font color="Maroon">q'</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span>;<br/>



<a NAME="E1:24_1_1"/>
( <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a>  <a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a>  <a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">p'</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">q'</font></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E2:24_1_1"/>
( (  not <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> or  not <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> or <font color="Maroon">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> ) &amp; (  not <font color="Maroon">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> or ( <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> ) ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:24_1"/>
<font color="Maroon">x</font> is    <a href="filter_2.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> 
 ;<br/>


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

<b>assume </b><a NAME="E2:24"/>
<font color="Maroon">x</font> is    <a href="filter_2.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">x</font> as    <a href="filter_2.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  ;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><a NAME="E1:24_2"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="filter_2.html#K5" target="_self">.:</a>  &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="filter_2.html#K5" target="_self">.:</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q</font> )
 ;<br/><b>hence </b><a NAME="E2:24_2"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> ) iff <font color="Maroon">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="filter_0.html#D1">FILTER_0:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:24"/>
<font color="Maroon">x</font> is    <a href="filter_2.html#M2" target="_self">Ideal</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="filter_2.html#T1" target="_self">Th1</a></i>;<br/>


</div>
