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<div class="add">

<b>let </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D1" target="_self" title="FILTER_0:def.1">FILTER_0:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font> ) iff <font color="Maroon" title="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font> )</span><br/>

<b>thus </b><a NAME="E1:48_1_1"/>
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font> implies <font color="Maroon" title="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font> implies ( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font> or <font color="Maroon" title="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">p2</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="E3:48_1_1_1"/><i><font color="Green" title="E36">A1</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">p2</font> &amp; <font color="Maroon" title="c6">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F2</font> )
 ;<br/>
<b>assume </b><a NAME="E4:48_1_1_1"/>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font></span><br/>

<b>then consider </b><font color="Maroon" title="c8">q1</font>, <font color="Maroon" title="c9">q2</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="E6:48_1_1_1"/><i><font color="Green" title="E37">A2</font></i>: 
( <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q2</font> &amp; <font color="Maroon" title="c8">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> &amp; <font color="Maroon" title="c9">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F2</font> )
 ;<br/>
<a NAME="E7:48_1_1_1"/><i><font color="Green" title="E38">A3</font></i>: 
( <font color="Maroon" title="c6">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F2</font> )
 
<b>by </b><i><a class="txt" href="#E3:48_1_1_1"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="#E6:48_1_1_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="filter_0.html#D1" target="_self" title="FILTER_0:def.1">Def1</a></i>;<br/>
<font color="Maroon" title="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">q</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">p2</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q2</font>
<b>by </b><i><a class="txt" href="#E3:48_1_1_1"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="#E6:48_1_1_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">p2</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q2</font>
<b>by </b><i><a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q2</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E9:48_1_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font>
 <b>by </b><i><a class="txt" href="#E7:48_1_1_1"><i><font color="Green" title="E38">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:48_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">q1</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="E4:48_1_1"/><i><font color="Green" title="E36">A4</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">q1</font> &amp; <font color="Maroon" title="c6">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> &amp; <font color="Maroon" title="c7">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F2</font> )
 ;<br/>
<a NAME="E5:48_1_1"/><i><font color="Green" title="E37">A5</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">q</font></span>)</span> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">q</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="lattices.html#D8" title="LATTICES:def.8">LATTICES:def 8</a></i>;<br/>
<a NAME="E6:48_1_1"/>
( <font color="Maroon" title="c4">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">q1</font></span>)</span> &amp; <font color="Maroon" title="c4">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> &amp; <font color="Maroon" title="c4">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F2</font> &amp; <font color="Maroon" title="c5">q</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">q1</font></span>)</span> &amp; <font color="Maroon" title="c5">q</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> &amp; <font color="Maroon" title="c5">q</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F2</font> )
 
<b>by </b><i><a class="txt" href="#E4:48_1_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="filter_0.html#T10" target="_self" title="FILTER_0:th.10">Th10</a>, <a class="ref" href="lattices.html#T31" title="LATTICES:th.31">LATTICES:31</a></i>;<br/>
<b>hence </b><a NAME="E7:48_1_1"/>
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">F2</font> )
 <b>by </b><i><a class="txt" href="#E4:48_1_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="#E5:48_1_1"><i><font color="Green" title="E37">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
