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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <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="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> ) &amp; ( <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><br/><a NAME="E1:58_1_1_1"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D8" title="LATTICES:def.8">LATTICES:def 8</a></span>;<br/><span class="kw">thus </span><a NAME="E2:58_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> &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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="58_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:58_1_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:58_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:58_1_1_1_1"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
( <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/>
<span class="kw">assume </span><a NAME="E5:58_1_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E7:58_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A4</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:58_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
( <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/>
<span class="lab"><font color="Green" title="E40">A6</font></span>: <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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1_1_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E7:58_1_1_1_1"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a></span>
;<br/>
<a NAME="E10:58_1_1_1_1"/>
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1_1"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E8:58_1_1_1_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="filter_0.html#T8" target="_self" title="FILTER_0:th.8">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:58_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:58_1_1_1_1"><span class="lab"><font color="Green" title="E40">A6</font></span></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/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E3:58_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>
 ; <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="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/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:58_1_1_1"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:58_1_1_1"/><span class="lab"><font color="Green" title="E37">A8</font></span>: 
( <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="E7:58_1_1_1"/><span class="lab"><font color="Green" title="E38">A9</font></span>: 
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_1_1_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="filter_0.html#T10" target="_self" title="FILTER_0:th.10">Th10</a></span>;<br/><a NAME="E8:58_1_1_1"/><span class="lab"><font color="Green" title="E39">A10</font></span>: 
( <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="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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T31" title="LATTICES:th.31">LATTICES:31</a></span>;<br/><a NAME="E9:58_1_1_1"/>
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_1_1_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="filter_0.html#T10" target="_self" title="FILTER_0:th.10">Th10</a></span>;<br/><span class="kw">hence </span><a NAME="E10:58_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> &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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E1:58_1_1_1"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E8:58_1_1_1"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E7:58_1_1_1"><span class="lab"><font color="Green" title="E38">A9</font></span></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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:58_1_1"/>
<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> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T8" target="_self" title="FILTER_0:th.8">Def1</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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