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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">F</font> be    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">H</font> be    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font>;<br/>





<a NAME="E1:54"/>
( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="filter_0.html#K5" target="_self">"/\"</a> <font color="Maroon">H</font> &amp; <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="filter_0.html#K5" target="_self">"/\"</a> <font color="Maroon">H</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:54"/><b>then </b>
<font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="filter_0.html#K5" target="_self">"/\"</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E3:54"/>
<span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H</font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="filter_0.html#K5" target="_self">"/\"</a> <font color="Maroon">H</font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<a NAME="E4:54"/>
<font color="Maroon">F</font> <a href="filter_0.html#K5" target="_self">"/\"</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H</font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:54_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="filter_0.html#K5" target="_self">"/\"</a> <font color="Maroon">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:54_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> &amp; <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">H</font> )
 ;<br/>
<a NAME="E4:54_1"/>
( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H</font> &amp; <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E5:54_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H</font> &amp; <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H</font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> &amp; <font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H</font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> )
 
<b>by </b><i><a class="ref" href="filter_0.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>hence </b><a NAME="E6:54_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H</font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span>
 <b>by </b><i><a class="ref" href="filter_0.html#D1" target="_self">Def1</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:54"/><b>then </b>
<span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="filter_0.html#K5" target="_self">"/\"</a> <font color="Maroon">H</font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H</font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span></span><a href="filter_0.html#K3" target="_self">.)</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:54"/>
<span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="filter_0.html#K5" target="_self">"/\"</a> <font color="Maroon">H</font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">H</font></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span>
 <b>by </b><i/>;<br/>


</div>
