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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  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c1">L</font> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> &amp; <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>,<span class="p1"><a href="filter_2.html#K18" title="FILTER_2:func.18">(.</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="filter_2.html#K18" title="FILTER_2:func.18">.&gt;</a></span> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a></span><br/><b>let </b><font color="Maroon" title="c2">p</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> &amp; <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>,<span class="p1"><a href="filter_2.html#K18" title="FILTER_2:func.18">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K18" title="FILTER_2:func.18">.&gt;</a></span> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> )</span><br/>



<b>assume </b><a NAME="E1:115"/><i><font color="Green" title="E49">A1</font></i>: 
( <font color="Maroon" title="c1">L</font> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V12" title="LATTICES:attr.12">modular</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>,<span class="p1"><a href="filter_2.html#K18" title="FILTER_2:func.18">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K18" title="FILTER_2:func.18">.&gt;</a></span> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a></span><br/>

<b>then reconsider </b><font color="Maroon" title="c3">K</font> =  <a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K18" title="FILTER_2:func.18">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K18" title="FILTER_2:func.18">.&gt;</a></span> as  <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> <b>by </b><i><a class="ref" href="filter_2.html#T83" target="_self" title="FILTER_2:th.83">Th83</a></i>;<br/>
<a NAME="E3:115"/>
<font color="Maroon" title="c3">K</font> is <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">b'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">K</font> st <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c4">b'</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">b</font> = <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>consider </b><font color="Maroon" title="c6">a</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="E3:115_1"/><i><font color="Green" title="E50">A2</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="filter_2.html#E1:115"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a></i>;<br/>
<a NAME="E4:115_1"/><i><font color="Green" title="E51">A3</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c6">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E3:115_1"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></i>;<br/>
<a NAME="E5:115_1"/><i><font color="Green" title="E52">A4</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">K</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K18" title="FILTER_2:func.18">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K18" title="FILTER_2:func.18">.&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="filter_2.html#T73" target="_self" title="FILTER_2:th.73">Th73</a>, <a class="ref" href="lattices.html#T23" title="LATTICES:th.23">LATTICES:23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c2">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font> <b>by </b><i><a class="ref" href="filter_2.html#T29" target="_self" title="FILTER_2:th.29">Th29</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">a'</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">a'</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c4">b'</font></span><br/>

<a NAME="E7:115_1"/><i><font color="Green" title="E53">A5</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="txt" href="filter_2.html#E5:115_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="filter_2.html#T29" target="_self" title="FILTER_2:th.29">Th29</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c7">a'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">b</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="c5">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="txt" href="filter_2.html#E1:115"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="filter_2.html#E7:115_1"><i><font color="Green" title="E53">A5</font></i></a>, <a class="ref" href="lattices.html#D12" title="LATTICES:def.12">LATTICES:def 12</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="txt" href="filter_2.html#E1:115"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="filter_2.html#E4:115_1"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="lattices.html#T43" title="LATTICES:th.43">LATTICES:43</a></i>
<br/>.= 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c3">K</font>
<b>by </b><i><a class="ref" href="filter_2.html#T81" target="_self" title="FILTER_2:th.81">Th81</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</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">b'</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c7">a'</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c4">b'</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">K</font> )</span><br/>

<b>hence </b><a NAME="E9:115_1"/>
<font color="Maroon" title="c4">b'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c3">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">a'</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c4">b'</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">K</font> )</span><br/>

<b>thus </b><font color="Maroon" title="c7">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">b'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">b</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="c2">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span>
<b>by </b><i><a class="txt" href="filter_2.html#E4:115_1"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a></i>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="txt" href="filter_2.html#E1:115"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="lattices.html#T40" title="LATTICES:th.40">LATTICES:40</a></i>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">K</font>
<b>by </b><i><a class="txt" href="filter_2.html#E1:115"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="filter_2.html#T82" target="_self" title="FILTER_2:th.82">Th82</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">b'</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">K</font></span><br/>

<b>hence </b><a NAME="E11:115_1"/>
<font color="Maroon" title="c4">b'</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">K</font>
 ; <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>hence </b><a NAME="E4:115"/>
 <a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K18" title="FILTER_2:func.18">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K18" title="FILTER_2:func.18">.&gt;</a></span> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
