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

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="lattices.html#L3" title="LATTICES:struct.3">LattStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> iff ( <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>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:36"/>
( <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>  ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:36_1"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">L9</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="E3:36_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">L9</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">L9</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="36_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c3">c</font> =  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c2">L9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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">L9</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> )</span><br/>

<span class="kw">thus </span><a NAME="E1:36_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">L9</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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T39" title="LATTICES:th.39">LATTICES:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:36_1"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c1">L</font> is  <a href="sheffer1.html#V2" title="SHEFFER1:attr.2">lower-bounded'</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sheffer1.html#D3" target="_self" title="SHEFFER1:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  )</span><br/>

<a NAME="E5:36_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">L9</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">L9</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="36_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c3">c</font> =  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c2">L9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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">L9</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> )</span><br/>

<span class="kw">thus </span><a NAME="E1:36_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">L9</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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T43" title="LATTICES:th.43">LATTICES:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:36_1"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon" title="c1">L</font> is  <a href="sheffer1.html#V1" title="SHEFFER1:attr.1">upper-bounded'</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sheffer1.html#D1" target="_self" title="SHEFFER1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  )</span><br/>

<span class="kw">thus </span><a NAME="E7:36_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  )</span><br/>

<a NAME="E8:36_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">L9</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a></span>;<br/>
<a NAME="E9:36_1"/><span class="kw">then </span>
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">L9</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T19" title="LATTICES:th.19">LATTICES:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:36_1"/>
( <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>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sheffer1.html#D5" target="_self" title="SHEFFER1:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">L</font> is  <a href="sheffer1.html#V4" title="SHEFFER1:attr.4">complemented'</a> </span><br/>

<span class="kw">hence </span><a NAME="E11:36_1"/>
<font color="Maroon" title="c1">L</font> is  <a href="sheffer1.html#V4" title="SHEFFER1:attr.4">complemented'</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E4:36_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E6:36_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="sheffer1.html#T25" target="_self" title="SHEFFER1:th.25">Th25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:36"/>
( <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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:36_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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">L9</font> = <font color="Maroon" title="c1">L</font> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="sheffer1.html#T10" target="_self" title="SHEFFER1:th.10">Th10</a></span>;<br/>
<a NAME="E3:36_2"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
<font color="Maroon" title="c2">L9</font> is  <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sheffer1.html#T24" target="_self" title="SHEFFER1:th.24">Th24</a></span>;<br/>
<a NAME="E4:36_2"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
( <font color="Maroon" title="c2">L9</font> is  <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a>  &amp; <font color="Maroon" title="c2">L9</font> is  <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sheffer1.html#T13" target="_self" title="SHEFFER1:th.13">Th13</a>, <a class="ref" href="sheffer1.html#T15" target="_self" title="SHEFFER1:th.15">Th15</a></span>;<br/>
<a NAME="E5:36_2"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
( <font color="Maroon" title="c2">L9</font> is  <a href="lattices.html#V8" title="LATTICES:attr.8">meet-absorbing</a>  &amp; <font color="Maroon" title="c2">L9</font> is  <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a>  )
 
<span class="kw">by</span> <span class="lab"><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></span>;<br/>
<a NAME="E6:36_2"/><span class="kw">then </span>
( <font color="Maroon" title="c2">L9</font> is  <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a>  &amp; <font color="Maroon" title="c2">L9</font> is  <a href="lattices.html#V7" title="LATTICES:attr.7">meet-associative</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sheffer1.html#T17" target="_self" title="SHEFFER1:th.17">Th17</a>, <a class="ref" href="sheffer1.html#T18" target="_self" title="SHEFFER1:th.18">Th18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:36_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_2"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E3:36_2"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E5:36_2"><span class="lab"><font color="Green" title="E36">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
