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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">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> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/>  for <font color="Olive" title="b2">F</font> being   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b2">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="latticea.html#K2" title="LATTICEA:func.2">PrimeFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> iff ( <font color="Olive" title="b2">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">F</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</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">  for <font color="Olive" title="b1">F</font> being   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="latticea.html#K2" title="LATTICEA:func.2">PrimeFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">a</font> iff ( <font color="Olive" title="b1">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</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">F</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font> be   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</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="c3">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="latticea.html#K2" title="LATTICEA:func.2">PrimeFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">a</font> iff ( <font color="Maroon" title="c3">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">F</font> ) )</span><br/>





<div><span class="kw">hereby </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="c3">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">F</font> implies <font color="Maroon" title="c3">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="latticea.html#K2" title="LATTICEA:func.2">PrimeFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">a</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:39_1"/>
<font color="Maroon" title="c3">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="latticea.html#K2" title="LATTICEA:func.2">PrimeFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">a</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="c3">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">F</font> )</span><br/><a NAME="E2:39_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">F0</font> being   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">F0</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Olive" title="b1">F0</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</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">F0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="latticea.html#T24" target="_self" onmouseover="rs('latticea/T24')" onmouseout="rh()">Th17</a></span>;<br/><span class="kw">hence </span><a NAME="E3:39_1"/>
( <font color="Maroon" title="c3">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">F</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"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:39"/>
( <font color="Maroon" title="c3">F</font> is  <a href="filter_0.html#V2" title="FILTER_0:attr.2">prime</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">F</font> implies <font color="Maroon" title="c3">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="latticea.html#K2" title="LATTICEA:func.2">PrimeFilters</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="latticea.html#T24" target="_self" onmouseover="rs('latticea/T24')" onmouseout="rh()">Th17</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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