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<b>let </b><font color="Maroon" title="c1">L</font> be   <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#V4" title="ROBBINS1:attr.4">Robbins</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</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="E1:112_1_1"/>
( <font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V4" title="ROBBINS1:attr.4">Robbins</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</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#V4" title="ROBBINS1:attr.4">Robbins</a> <a href="robbins1.html#V9" title="ROBBINS1:attr.9">de_Morgan</a> <a href="robbins1.html#NM1" title="ROBBINS1:NM.1">preOrthoLattice</a> ;<br/>
<b>thus </b><a NAME="E3:112_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="112_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:112_1_1_1"/>
<font color="Maroon" title="c2">L'</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>
 
;<br/>
<b>hence </b><a NAME="E2:112_1_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a>
 ; <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><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_2"><b>now </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="c2">L'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L'</font> st <br/>( <font color="Olive" title="b1">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="c2">L'</font> &amp; <font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</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="c2">L'</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="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="c2">L'</font> &amp; <font color="Maroon" title="c4">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</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="c2">L'</font> )</span><br/><b>thus </b><a NAME="E1:112_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="c2">L'</font>
 <b>by </b><i><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#K4" title="LATTICES:func.4">"/\"</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="c2">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="c2">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"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:112_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><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_3"><b>now </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="c2">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="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> <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#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></span><br/><a NAME="E1:112_1_1_3"/><i><font color="Green" title="E62">A1</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="robbins1.html#E1:112_1_1_3"><i><font color="Green" title="E62">A1</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>
<b>hence </b><a NAME="E7:112_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><br/>


</div>
