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

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="lattices.html#L3" title="LATTICES:struct.3">LattStr</a> ;<br/>

<b>thus </b><a NAME="E1:35"/>
( <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> implies ( <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V2" title="SHEFFER1:attr.2">lower-bounded'</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V1" title="SHEFFER1:attr.1">upper-bounded'</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V3" title="SHEFFER1:attr.3">distributive'</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V4" title="SHEFFER1:attr.4">complemented'</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c2">L'</font> = <font color="Maroon" title="c1">L</font> as  <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> ;<br/>
<a NAME="E3:35_1"/>
 ex <font color="Olive" title="b1">c</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/>for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L'</font> holds <br/> ( <font color="Olive" title="b1">c</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c3">c</font> =  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c2">L'</font>;<br/>

<b>thus </b><a NAME="E1:35_1_1"/>
for <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> holds <br/> ( <font color="Maroon" title="c3">c</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> )
 <b>by </b><i><a class="ref" href="lattices.html#T39" title="LATTICES:th.39">LATTICES:39</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:35_1"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V2" title="SHEFFER1:attr.2">lower-bounded'</a>
 <b>by </b><i><a class="ref" href="sheffer1.html#D3" target="_self" title="SHEFFER1:def.3">Def3</a></i>;<br/>

<a NAME="E5:35_1"/>
 ex <font color="Olive" title="b1">c</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/>for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L'</font> holds <br/> ( <font color="Olive" title="b1">c</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c3">c</font> =  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c2">L'</font>;<br/>

<b>thus </b><a NAME="E1:35_1_2"/>
for <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> holds <br/> ( <font color="Maroon" title="c3">c</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> )
 <b>by </b><i><a class="ref" href="lattices.html#T43" title="LATTICES:th.43">LATTICES:43</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:35_1"/><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V1" title="SHEFFER1:attr.1">upper-bounded'</a>
 <b>by </b><i><a class="ref" href="sheffer1.html#D1" target="_self" title="SHEFFER1:def.1">Def1</a></i>;<br/>

<b>thus </b><a NAME="E7:35_1"/>
( <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a> )
 <b>by </b><i><a class="txt" href="sheffer1.html#E1:35_1"><i><font color="Green" title="E34">A1</font></i></a></i>;<br/>

<a NAME="E8:35_1"/>
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="c2">L'</font> holds  <font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</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#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a></i>;<br/>
<a NAME="E9:35_1"/><b>then </b>
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="c2">L'</font> holds  <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> <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="b3">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattices.html#T19" title="LATTICES:th.19">LATTICES:19</a></i>;<br/>
<b>hence </b><a NAME="E10:35_1"/><i><font color="Green" title="E37">A4</font></i>: 
( <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V3" title="SHEFFER1:attr.3">distributive'</a> )
 <b>by </b><i><a class="ref" href="sheffer1.html#D5" target="_self" title="SHEFFER1:def.5">Def5</a></i>;<br/>

<b>thus </b><a NAME="E11:35_1"/>
<font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V4" title="SHEFFER1:attr.4">complemented'</a>
 <b>by </b><i><a class="txt" href="sheffer1.html#E1:35_1"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="sheffer1.html#E4:35_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="sheffer1.html#E6:35_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="sheffer1.html#E10:35_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="sheffer1.html#T25" target="_self" title="SHEFFER1:th.25">Th25</a></i>;<br/>


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

<b>thus </b><a NAME="E2:35"/>
( <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V2" title="SHEFFER1:attr.2">lower-bounded'</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V1" title="SHEFFER1:attr.1">upper-bounded'</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V3" title="SHEFFER1:attr.3">distributive'</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V4" title="SHEFFER1:attr.4">complemented'</a> implies <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_2"/>
( <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V2" title="SHEFFER1:attr.2">lower-bounded'</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V1" title="SHEFFER1:attr.1">upper-bounded'</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V3" title="SHEFFER1:attr.3">distributive'</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V4" title="SHEFFER1:attr.4">complemented'</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c2">L'</font> = <font color="Maroon" title="c1">L</font> as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a> <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a> <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a> <a href="robbins1.html#V6" title="ROBBINS1:attr.6">join-idempotent</a> <a href="sheffer1.html#V1" title="SHEFFER1:attr.1">upper-bounded'</a> <a href="sheffer1.html#V2" title="SHEFFER1:attr.2">lower-bounded'</a> <a href="sheffer1.html#V3" title="SHEFFER1:attr.3">distributive'</a> <a href="sheffer1.html#V4" title="SHEFFER1:attr.4">complemented'</a>  <a href="lattices.html#L3" title="LATTICES:struct.3">LattStr</a>  <b>by </b><i><a class="ref" href="sheffer1.html#T10" target="_self" title="SHEFFER1:th.10">Th10</a></i>;<br/>
<a NAME="E3:35_2"/><i><font color="Green" title="E34">A5</font></i>: 
<font color="Maroon" title="c2">L'</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a>
 
<b>by </b><i><a class="ref" href="sheffer1.html#T15" target="_self" title="SHEFFER1:th.15">Th15</a></i>;<br/>
<a NAME="E4:35_2"/>
<font color="Maroon" title="c2">L'</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>
 
<b>by </b><i><a class="ref" href="sheffer1.html#T13" target="_self" title="SHEFFER1:th.13">Th13</a></i>;<br/>
<a NAME="E5:35_2"/><b>then </b><i><font color="Green" title="E35">A6</font></i>: 
<font color="Maroon" title="c2">L'</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a>
 
<b>by </b><i><a class="txt" href="sheffer1.html#E3:35_2"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="lattices.html#D15" title="LATTICES:def.15">LATTICES:def 15</a></i>;<br/>
<a NAME="E6:35_2"/><i><font color="Green" title="E36">A7</font></i>: 
<font color="Maroon" title="c2">L'</font> is <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a>
 
<b>by </b><i><a class="ref" href="sheffer1.html#T24" target="_self" title="SHEFFER1:th.24">Th24</a></i>;<br/>
<a NAME="E7:35_2"/><i><font color="Green" title="E37">A8</font></i>: 
( <font color="Maroon" title="c2">L'</font> is <a href="lattices.html#V8" title="LATTICES:attr.8">meet-absorbing</a> &amp; <font color="Maroon" title="c2">L'</font> is <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a> )
 
<b>by </b><i><a class="ref" href="sheffer1.html#T11" target="_self" title="SHEFFER1:th.11">Th11</a>, <a class="ref" href="sheffer1.html#T12" target="_self" title="SHEFFER1:th.12">Th12</a></i>;<br/>
<a NAME="E8:35_2"/><b>then </b><i><font color="Green" title="E38">A9</font></i>: 
<font color="Maroon" title="c2">L'</font> is <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a>
 
<b>by </b><i><a class="ref" href="sheffer1.html#T17" target="_self" title="SHEFFER1:th.17">Th17</a></i>;<br/>
<a NAME="E9:35_2"/>
<font color="Maroon" title="c2">L'</font> is <a href="lattices.html#V7" title="LATTICES:attr.7">meet-associative</a>
 
<b>by </b><i><a class="txt" href="sheffer1.html#E7:35_2"><i><font color="Green" title="E37">A8</font></i></a>, <a class="ref" href="sheffer1.html#T18" target="_self" title="SHEFFER1:th.18">Th18</a></i>;<br/>
<b>hence </b><a NAME="E10:35_2"/>
<font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>
 <b>by </b><i><a class="txt" href="sheffer1.html#E5:35_2"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="sheffer1.html#E6:35_2"><i><font color="Green" title="E36">A7</font></i></a>, <a class="txt" href="sheffer1.html#E7:35_2"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="sheffer1.html#E8:35_2"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="lattices.html#D10" title="LATTICES:def.10">LATTICES:def 10</a>, <a class="ref" href="lattices.html#D20" title="LATTICES:def.20">LATTICES:def 20</a></i>;<br/>


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


</div>
