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

<b>let </b><font color="Maroon">L</font> be  <a href="robbins1.html#V9" target="_self">de_Morgan</a> <a href="robbins1.html#NM1">preOrthoLattice</a>;<br/>

<b>assume </b><a NAME="E1:111_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">L</font> is <a href="robbins1.html#V5" target="_self">Huntington</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">L'</font> = <font color="Maroon">L</font> as  <a href="robbins1.html#V5" target="_self">Huntington</a> <a href="robbins1.html#NM1">preOrthoLattice</a> ;<br/>
<b>thus </b><a NAME="E3:111_1_1"/>
<font color="Maroon">L</font> is <a href="lattices.html#V15">bounded</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D20">LATTICES:def 20</a><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">E35</font></i>: 
<font color="Maroon">L'</font> is <a href="lattices.html#V14">upper-bounded</a>
 
;<br/>
<a NAME="E2:111_1_1_1"/>
<font color="Maroon">L</font> is <a href="lattices.html#V13">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">c'</font> =  <a href="robbins1.html#K7" target="_self">Bot</a> <font color="Maroon">L'</font>;<br/>
<b>reconsider </b><font color="Maroon">c</font> =  <a href="robbins1.html#K7" target="_self">Bot</a> <font color="Maroon">L'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>take </b>
<font color="Maroon">c</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D13">LATTICES:def 13</a><br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> ;<br/>
<b>thus </b><font color="Maroon">c</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><a href="robbins1.html#K7" target="_self">Bot</a> <font color="Maroon">L'</font></span>)</span> <a href="robbins1.html#K6" target="_self">*'</a> <font color="Maroon">a'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">c</font>
<b>by </b><i/>
;<br/>

<b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><a href="robbins1.html#K7" target="_self">Bot</a> <font color="Maroon">L'</font></span>)</span> <a href="robbins1.html#K6" target="_self">*'</a> <font color="Maroon">a'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">c</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:111_1_1_1"/>
<font color="Maroon">L</font> is <a href="lattices.html#V15">bounded</a>
 <b>by </b><i><a class="ref" href="robbins1.html#T28" target="_self">Th28</a>, <a class="ref" href="lattices.html#D15">LATTICES:def 15</a></i>;<br/>


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

<b>thus </b><a NAME="E4:111_1_1"/>
<font color="Maroon">L</font> is <a href="lattices.html#V16">complemented</a>
 <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">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19">LATTICES:def 19</a><br/>

<b>take </b><font color="Maroon">a</font> = <font color="Maroon">b</font> <a href="robbins1.html#K3" target="_self">`</a> ;<br/>

<a NAME="E1:111_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">L'</font> is <a href="robbins1.html#V6" target="_self">join-idempotent</a>
 
;<br/>
<b>hence </b><a NAME="E2:111_1_1_2"/>
<font color="Maroon">a</font> <a href="robbins1.html#K5" target="_self">+</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D18">LATTICES:def 18</a><br/>

<b>thus </b><a NAME="E3:111_1_1_2"/>
<font color="Maroon">b</font> <a href="robbins1.html#K5" target="_self">+</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="robbins1.html#T28" target="_self">Th28</a>, </i>;<br/>

<b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> = 
<font color="Maroon">a</font> <a href="robbins1.html#K6" target="_self">*'</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> 

<br/>.= 
 <a href="robbins1.html#K7" target="_self">Bot</a> <font color="Maroon">L'</font>
<b>by </b><i/>
<br/>.= 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="robbins1.html#T26" target="_self">Th26</a></i>
;<br/>

<b>hence </b><a NAME="E5:111_1_1_2"/>
<font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font>
 ;<br/>


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

<b>thus </b><a NAME="E5:111_1_1"/>
<font color="Maroon">L</font> is <a href="lattices.html#V11">distributive</a>
 <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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D11">LATTICES:def 11</a><br/>

<a NAME="E1:111_1_1_3"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="robbins1.html#K6" target="_self">*'</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="robbins1.html#K6" target="_self">*'</a> <font color="Maroon">c</font> )
 
<b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c</font></span>)</span> = 
<font color="Maroon">a</font> <a href="robbins1.html#K6" target="_self">*'</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K5" target="_self">+</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#T27" target="_self">Th27</a>, <a class="ref" href="robbins1.html#T28" target="_self">Th28</a>, <a class="ref" href="robbins1.html#D8" target="_self">Def8</a></i>
;<br/>


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


</div>
