<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</a> <a href="robbins1.html#NM1" title="ROBBINS1:NM.1">preOrthoLattice</a>; <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="robbins1.html#V5" title="ROBBINS1:attr.5">Huntington</a> implies <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a> )</span><br/>

<b>assume </b><a NAME="E1:111_1_1"/><i><font color="Green" title="E62">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V5" title="ROBBINS1:attr.5">Huntington</a>
 ; <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#V17" title="LATTICES:attr.17">Boolean</a></span><br/>

<b>then reconsider </b><font color="Maroon" title="c2">L'</font> = <font color="Maroon" title="c1">L</font> as  <a href="robbins1.html#V5" title="ROBBINS1:attr.5">Huntington</a> <a href="robbins1.html#NM1" title="ROBBINS1:NM.1">preOrthoLattice</a> ;<br/>
<b>thus </b><a NAME="E3:111_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D20" title="LATTICES:def.20">LATTICES:def 20</a> <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#V16" title="LATTICES:attr.16">complemented</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:111_1_1_1"/><i><font color="Green" title="E63">A2</font></i>: 
<font color="Maroon" title="c2">L'</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>
 
;<br/>
<a NAME="E2:111_1_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c3">c'</font> =  <a href="robbins1.html#K7" title="ROBBINS1:func.7">Bot</a> <font color="Maroon" title="c2">L'</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">c</font> =  <a href="robbins1.html#K7" title="ROBBINS1:func.7">Bot</a> <font color="Maroon" title="c2">L'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c4">c</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D13" title="LATTICES:def.13">LATTICES:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>) holds <br/> ( <font color="Maroon" title="c4">c</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> )</span><br/>

<b>let </b><font color="Maroon" title="c5">a</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="c4">c</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c5">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c6">a'</font> = <font color="Maroon" title="c5">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L'</font> ;<br/>
<b>thus </b><font color="Maroon" title="c4">c</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">a</font> = 
<span class="p1">(<span class="default"><a href="robbins1.html#K7" title="ROBBINS1:func.7">Bot</a> <font color="Maroon" title="c2">L'</font></span>)</span> <a href="robbins1.html#K6" title="ROBBINS1:func.6">*'</a> <font color="Maroon" title="c6">a'</font>
<b>by </b><i><a class="ref" href="robbins1.html#D23" target="_self" title="ROBBINS1:def.23">Def23</a></i>
<br/>.= 
<font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="ref" href="robbins1.html#D9" target="_self" title="ROBBINS1:def.9">Def9</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="c5">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font></span><br/>

<b>thus </b><font color="Maroon" title="c5">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">c</font> = 
<span class="p1">(<span class="default"><a href="robbins1.html#K7" title="ROBBINS1:func.7">Bot</a> <font color="Maroon" title="c2">L'</font></span>)</span> <a href="robbins1.html#K6" title="ROBBINS1:func.6">*'</a> <font color="Maroon" title="c6">a'</font>
<b>by </b><i><a class="ref" href="robbins1.html#D23" target="_self" title="ROBBINS1:def.23">Def23</a></i>
<br/>.= 
<font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="ref" href="robbins1.html#D9" target="_self" title="ROBBINS1:def.9">Def9</a></i>
; <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="E3:111_1_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a>
 <b>by </b><i><a class="txt" href="#E1:111_1_1_1"><i><font color="Green" title="E63">A2</font></i></a></i>; <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>thus </b><a NAME="E4:111_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a>
  <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#V11" title="LATTICES:attr.11">distributive</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</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">M2</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</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="c3">b</font></span><br/>

<b>take </b><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c3">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</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">a</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c3">b</font></span><br/>

<a NAME="E1:111_1_1_2"/><i><font color="Green" title="E63">A3</font></i>: 
<font color="Maroon" title="c2">L'</font> is <a href="robbins1.html#V6" title="ROBBINS1:attr.6">join-idempotent</a>
 
;<br/>
<b>hence </b><a NAME="E2:111_1_1_2"/>
<font color="Maroon" title="c4">a</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c3">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>
 <b>by </b><i><a class="ref" href="robbins1.html#D8" target="_self" title="ROBBINS1:def.8">Def8</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="c3">b</font> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <font color="Maroon" title="c4">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="c1">L</font> &amp; <font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">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> &amp; <font color="Maroon" title="c3">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">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="c1">L</font> )</span><br/>

<b>thus </b><a NAME="E3:111_1_1_2"/>
<font color="Maroon" title="c3">b</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c4">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="c1">L</font>
 <b>by </b><i><a class="txt" href="#E1:111_1_1_2"><i><font color="Green" title="E63">A3</font></i></a>, <a class="ref" href="robbins1.html#D8" target="_self" title="ROBBINS1:def.8">Def8</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">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">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> &amp; <font color="Maroon" title="c3">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">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="c1">L</font> )</span><br/>

<b>thus </b><font color="Maroon" title="c4">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" title="ROBBINS1:func.3">`</a> 
<b>by </b><i><a class="ref" href="robbins1.html#D23" target="_self" title="ROBBINS1:def.23">Def23</a></i>
<br/>.= 
 <a href="robbins1.html#K7" title="ROBBINS1:func.7">Bot</a> <font color="Maroon" title="c2">L'</font>
<b>by </b><i><a class="ref" href="robbins1.html#T9" target="_self" title="ROBBINS1:th.9">Th9</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="ref" href="robbins1.html#T63" target="_self" title="ROBBINS1:th.63">Th63</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="c3">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">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="c1">L</font></span><br/>

<b>hence </b><a NAME="E5:111_1_1_2"/>
<font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">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="c1">L</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>thus </b><a NAME="E5:111_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_1_3"><b>proof </b></a><div class="add">

<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 <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a> <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#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="robbins1.html#NK4" title="ROBBINS1:NK.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#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="robbins1.html#NK4" title="ROBBINS1:NK.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">c</font></span>)</span></span><br/>

<a NAME="E1:111_1_1_3"/><i><font color="Green" title="E63">A4</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="robbins1.html#K6" title="ROBBINS1:func.6">*'</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c3">a</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="c3">a</font> <a href="robbins1.html#K6" title="ROBBINS1:func.6">*'</a> <font color="Maroon" title="c5">c</font> )
 
<b>by </b><i><a class="ref" href="robbins1.html#D23" target="_self" title="ROBBINS1:def.23">Def23</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">c</font></span>)</span> = 
<font color="Maroon" title="c3">a</font> <a href="robbins1.html#K6" title="ROBBINS1:func.6">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="robbins1.html#K5" title="ROBBINS1:func.5">+</a> <font color="Maroon" title="c5">c</font></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#D23" target="_self" title="ROBBINS1:def.23">Def23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">c</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:111_1_1"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="#E1:111_1_1_3"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="robbins1.html#T31" target="_self" title="ROBBINS1:th.31">Th31</a></i>
; <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>


</div>
