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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>  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_0.html#K8" title="FILTER_0:func.8">latt</a> <font color="Olive" title="b1">F</font> is  <a href="lattices.html#V12" title="LATTICES:attr.12">modular</a> </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"> ( <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V12" title="LATTICES:attr.12">modular</a>  implies  <a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <font color="Maroon" title="c2">F</font> is  <a href="lattices.html#V12" title="LATTICES:attr.12">modular</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:82"/><span class="lab"><font color="Green" title="E50">A1</font></span>: 
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>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D12" title="LATTICES:def.12">LATTICES:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <font color="Maroon" title="c2">F</font> is  <a href="lattices.html#V12" title="LATTICES:attr.12">modular</a> </span><br/>

<span class="kw">let </span><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_0.html#K8" title="FILTER_0:func.8">latt</a> <font color="Maroon" title="c2">F</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D12" title="LATTICES:def.12">LATTICES:def 12</a> <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="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/>

<span class="kw">assume </span><a NAME="E2:82"/><span class="lab"><font color="Green" title="E51">A2</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c7">q</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c5">c</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T63" target="_self" title="FILTER_0:th.63">Th63</a></span>;<br/>
<a NAME="E4:82"/>
<font color="Maroon" title="c4">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T64" target="_self" title="FILTER_0:th.64">Th64</a></span>;<br/>
<a NAME="E5:82"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A3</font></span>: 
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T64" target="_self" title="FILTER_0:th.64">Th64</a></span>;<br/>
<a NAME="E6:82"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T64" target="_self" title="FILTER_0:th.64">Th64</a></span>;<br/>
<a NAME="E7:82"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A4</font></span>: 
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">q</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T64" target="_self" title="FILTER_0:th.64">Th64</a></span>;<br/>
<a NAME="E8:82"/>
<font color="Maroon" title="c6">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="ref" href="filter_0.html#T65" target="_self" title="FILTER_0:th.65">Th65</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:82"/>
<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> <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#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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="txt" href="#E7:82"><span class="lab"><font color="Green" title="E53">A4</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/>


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