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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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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><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/><span class="kw">let </span><font color="Maroon" title="c2">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>







<a NAME="E1:6"/><span class="lab"><font color="Green" title="E1">A1</font></span>: 
<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">p9</font>, <font color="Olive" title="b3">q9</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">p9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">q9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b2">p9</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">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T48" title="FILTER_0:th.48">FILTER_0:48</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:6"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E3:6"/><span class="kw">then </span>
 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">p9</font>, <font color="Olive" title="b3">q9</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">p9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">q9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b2">p9</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">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E1">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">p9</font>, <font color="Maroon" title="c6">q9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E5:6"/><span class="lab"><font color="Green" title="E2">A2</font></span>: 
<font color="Maroon" title="c5">p9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">q9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">p</font>
 <span class="kw">and </span><br/><a NAME="E6:6"/><span class="lab"><font color="Green" title="E3">A3</font></span>: 
<font color="Maroon" title="c5">p9</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>
 <span class="kw">and </span><br/><a NAME="E7:6"/><span class="lab"><font color="Green" title="E4">A4</font></span>: 
<font color="Maroon" title="c6">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ;<br/>
<a NAME="E8:6"/>
<font color="Maroon" title="c4">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:6"><span class="lab"><font color="Green" title="E3">A3</font></span></a>, <a class="ref" href="filter_0.html#T18" title="FILTER_0:th.18">FILTER_0:18</a></span>;<br/>
<a NAME="E9:6"/><span class="kw">then </span>
<font color="Maroon" title="c4">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">q9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">p9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">q9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T27" title="LATTICES:th.27">LATTICES:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:6"/>
 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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="txt" href="#E7:6"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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