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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>;<br/><b>let </b><font color="Maroon" title="c2">P</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="lattice4.html#M2" title="LATTICE4:mode.2">ClosedSubset</a> of <font color="Maroon" title="c1">L</font>;<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>;<br/><b>let </b><font color="Maroon" title="c5">p'</font>, <font color="Maroon" title="c6">q'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">P</font></span>)</span>;<br/>









<b>assume </b><a NAME="E1:105"/><i><font color="Green" title="E44">A1</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p'</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q'</font> )
 ;<br/>

<b>thus </b><a NAME="E2:105"/>
( <font color="Maroon" title="c3">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">q</font> implies <font color="Maroon" title="c5">p'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:105_1"/>
<font color="Maroon" title="c3">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a><br/>

<b>hence </b><a NAME="E2:105_1"/>
<font color="Maroon" title="c5">p'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q'</font>
 <b>by </b><i><a class="txt" href="filter_2.html#E1:105"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="filter_2.html#T74" target="_self" title="FILTER_2:th.74">Th74</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a><br/>


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

<b>assume </b><a NAME="E3:105"/>
<font color="Maroon" title="c5">p'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q'</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a><br/>

<b>hence </b><a NAME="E4:105"/>
<font color="Maroon" title="c3">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="filter_2.html#E1:105"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="filter_2.html#T74" target="_self" title="FILTER_2:th.74">Th74</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a><br/>


</div>
