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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">P</font> being 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>  st <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V12" title="LATTICES:attr.12">modular</a> holds <br/> <a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<font color="Olive" title="b1">P</font> is <a href="lattices.html#V12" title="LATTICES:attr.12">modular</a></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V12" title="LATTICES:attr.12">modular</a> implies  <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> is <a href="lattices.html#V12" title="LATTICES:attr.12">modular</a> )</span><br/>



<b>assume </b><a NAME="E1:107"/><i><font color="Green" title="E43">A1</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</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">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b3">c</font> holds <br/><font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">c</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D12" title="LATTICES:def.12">LATTICES:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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> is <a href="lattices.html#V12" title="LATTICES:attr.12">modular</a></span><br/>

<b>let </b><font color="Maroon" title="c3">a'</font>, <font color="Maroon" title="c4">b'</font>, <font color="Maroon" title="c5">c'</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D12" title="LATTICES:def.12">LATTICES:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">a'</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c5">c'</font> or <font color="Maroon" title="c3">a'</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b'</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">c'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a'</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">b'</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">c'</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c3">a'</font>, <font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c4">b'</font>, <font color="Maroon" title="c8">c</font> = <font color="Maroon" title="c5">c'</font>, <font color="Maroon" title="c9">bc</font> = <font color="Maroon" title="c4">b'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">c'</font>, <font color="Maroon" title="c10">ab</font> = <font color="Maroon" title="c3">a'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="filter_2.html#T69" target="_self" title="FILTER_2:th.69">Th69</a></i>;<br/>
<b>assume </b><a NAME="E3:107"/>
<font color="Maroon" title="c3">a'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">c'</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">a'</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b'</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">c'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a'</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">b'</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">c'</font></span><br/>

<a NAME="E4:107"/><b>then </b><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">c</font>
 
<b>by </b><i><a class="ref" href="filter_2.html#T75" target="_self" title="FILTER_2:th.75">Th75</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">a'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">c'</font></span>)</span> = 
<font color="Maroon" title="c6">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c9">bc</font>
<b>by </b><i><a class="ref" href="filter_2.html#T74" target="_self" title="FILTER_2:th.74">Th74</a></i>
<br/>.= 
<font color="Maroon" title="c6">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">c</font></span>)</span>
<b>by </b><i><a class="ref" href="filter_2.html#T74" target="_self" title="FILTER_2:th.74">Th74</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">c</font>
<b>by </b><i><a class="txt" href="#E1:107"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="#E4:107"><i><font color="Green" title="E44">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">ab</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">c</font>
<b>by </b><i><a class="ref" href="filter_2.html#T74" target="_self" title="FILTER_2:th.74">Th74</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b'</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">c'</font>
<b>by </b><i><a class="ref" href="filter_2.html#T74" target="_self" title="FILTER_2:th.74">Th74</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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