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<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">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">F</font> being   <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">p</font> <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"><span class="p3"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Olive" title="b3">q</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b1">F</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> holds <br/> ex <font color="Olive" title="b4">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b3">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b4">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">p</font> )</span><br/><b>let </b><font color="Maroon" title="c2">F</font> be    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">L</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">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">p</font> <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"><span class="p3"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Olive" title="b2">q</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">F</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> holds <br/> ex <font color="Olive" title="b3">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b2">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">p</font> )</span><br/><b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">p</font> <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"><span class="p3"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c4">q</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">F</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> implies  ex <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c4">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">p</font> ) )</span><br/>







<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E1">A1</font></i>: 
<font color="Maroon" title="c3">p</font> <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"><span class="p3"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c4">q</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">F</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c4">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">p</font> )</span><br/>

<a NAME="E2:4"/>
<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"><span class="p3"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c4">q</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">F</font></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> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">p'</font>, <font color="Olive" title="b3">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">q'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b2">p'</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="c4">q</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> &amp; <font color="Olive" title="b3">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="filter_0.html#T48" title="FILTER_0:th.48">FILTER_0:48</a></i>;<br/>
<a NAME="E3:4"/><b>then </b>
 ex <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> &amp;  ex <font color="Olive" title="b2">p'</font>, <font color="Olive" title="b3">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">q'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b2">p'</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="c4">q</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> &amp; <font color="Olive" title="b3">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> ) )
 
<b>by </b><i><a class="txt" href="#E1:4"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">p'</font>, <font color="Maroon" title="c6">q'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:4"/><i><font color="Green" title="E2">A2</font></i>: 
( <font color="Maroon" title="c5">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">q'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Maroon" title="c5">p'</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="c4">q</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> &amp; <font color="Maroon" title="c6">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> )
 ;<br/>
<a NAME="E6:4"/>
<font color="Maroon" title="c4">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">p'</font>
 
<b>by </b><i><a class="txt" href="#E5:4"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="filter_0.html#T18" title="FILTER_0:th.18">FILTER_0:18</a></i>;<br/>
<a NAME="E7:4"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">q'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">q'</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T27" title="LATTICES:th.27">LATTICES:27</a></i>;<br/>
<b>hence </b><a NAME="E8:4"/>
 ex <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c4">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">p</font> )
 <b>by </b><i><a class="txt" href="#E5:4"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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