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

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="sheffer1.html#V9" target="_self">properly_defined</a> <a href="sheffer1.html#V10" target="_self">satisfying_Sheffer_1</a> <a href="sheffer1.html#V11" target="_self">satisfying_Sheffer_2</a> <a href="sheffer1.html#V12" target="_self">satisfying_Sheffer_3</a>  <a href="sheffer1.html#L3" target="_self">ShefferOrthoLattStr</a> ;<br/>

<a NAME="E1:60"/>
 ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <font color="Olive">c</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>
<b>take </b><font color="Maroon">c</font> = <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span>;<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>thus </b><a NAME="E2:60_1"/>
<font color="Maroon">c</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>proof </b></a><div class="add">

<font color="Maroon">c</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sheffer1.html#E6:60"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:60_1_1"/>
<font color="Maroon">c</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>


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

<b>thus </b><a NAME="E3:60_1"/>
<font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2"><b>proof </b></a><div class="add">

<font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:60_1_2"/>
<font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>


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


</div><b>end;</b></div>
<a NAME="E2:60"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">L</font> is <a href="sheffer1.html#V2" target="_self">lower-bounded'</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:60"/>
 ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <font color="Olive">c</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>
<b>take </b><font color="Maroon">c</font> = <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span>;<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>thus </b><a NAME="E2:60_2"/>
<font color="Maroon">c</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">X</font> = <font color="Maroon">b</font> <a href="sheffer1.html#K6" target="_self">"</a> ;<br/>
<font color="Maroon">c</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> 
<b>by </b><i><a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> 
<b>by </b><i><a class="txt" href="sheffer1.html#E6:60"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> 
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:60_2_1"/>
<font color="Maroon">c</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>


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

<b>thus </b><a NAME="E3:60_2"/>
<font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">X</font> = <font color="Maroon">b</font> <a href="sheffer1.html#K6" target="_self">"</a> ;<br/>
<font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> 
<b>by </b><i><a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> 
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:60_2_2"/>
<font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>


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


</div><b>end;</b></div>
<a NAME="E4:60"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">L</font> is <a href="sheffer1.html#V1" target="_self">upper-bounded'</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:60"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Olive">a</font> <a href="sheffer1.html#R1" target="_self">is_a_complement'_of</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3"><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>;<br/>

<a NAME="E1:60_3"/><i><font color="Green">E27</font></i>: 
 <a href="sheffer1.html#K1" target="_self">Top'</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">X</font> = <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:60_3_1"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_1_1"><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>;<br/>

<b>thus </b><a NAME="E1:60_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">Y</font> = <font color="Maroon">b</font> <a href="sheffer1.html#K6" target="_self">"</a> ;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> 
<b>by </b><i><a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> 
<b>by </b><i><a class="txt" href="sheffer1.html#E6:60"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> 
<b>by </b><i><a class="txt" href="sheffer1.html#E2:60_3"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:60_3_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>


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

<b>thus </b><a NAME="E2:60_3_1_1"/>
<font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_1_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">Y</font> = <font color="Maroon">b</font> <a href="sheffer1.html#K6" target="_self">"</a> ;<br/>
<font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> 
<b>by </b><i><a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> 
<b>by </b><i><a class="txt" href="sheffer1.html#E2:60_3"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:60_3_1_1_2"/>
<font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>


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

<b>thus </b><a NAME="E3:60_3_1_1"/>
verum
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:60_3_1"/>
 <a href="sheffer1.html#K1" target="_self">Top'</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sheffer1.html#T8" target="_self">Th8</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:60_3"/><i><font color="Green">E36</font></i>: 
 <a href="sheffer1.html#K2" target="_self">Bot'</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">X</font> = <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:60_3_2"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_2_1"><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>;<br/>

<b>thus </b><a NAME="E1:60_3_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_2_1_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="txt" href="sheffer1.html#E5:60_3"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sheffer1.html#E6:60"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="txt" href="sheffer1.html#E2:60_3"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:60_3_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>


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

<b>thus </b><a NAME="E2:60_3_2_1"/>
<font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_2_1_2"><b>proof </b></a><div class="add">

<font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span> <a href="sheffer1.html#K6" target="_self">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sheffer1.html#E5:60_3"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="txt" href="sheffer1.html#E2:60_3"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:60_3_2_1_2"/>
<font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>


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

<b>thus </b><a NAME="E3:60_3_2_1"/>
verum
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:60_3_2"/>
 <a href="sheffer1.html#K2" target="_self">Bot'</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">a</font> = <font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font>;<br/>
<b>take </b>
<font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font>
;<br/>

<i><font color="Green">E37</font></i>: <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>
<br/>.= 
 <a href="sheffer1.html#K1" target="_self">Top'</a> <font color="Maroon">L</font>
<b>by </b><i>, <a class="txt" href="sheffer1.html#E6:60"><i><font color="Green">E40</font></i></a></i>
;<br/>
<a NAME="E4:60_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">b</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sheffer1.html#K1" target="_self">Top'</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>;<br/>
<i><font color="Green">E39</font></i>: <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="txt" href="sheffer1.html#E6:60"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
 <a href="sheffer1.html#K2" target="_self">Bot'</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="sheffer1.html#D9" target="_self">Def9</a>, <a class="txt" href="sheffer1.html#E6:60"><i><font color="Green">E40</font></i></a></i>
;<br/>
<a NAME="E6:60_3"/>
<font color="Maroon">b</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sheffer1.html#K2" target="_self">Bot'</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="sheffer1.html#D9" target="_self">Def9</a>, <a class="ref" href="sheffer1.html#D8" target="_self">Def8</a></i>;<br/>
<b>hence </b><a NAME="E7:60_3"/>
<font color="Maroon">b</font> <a href="sheffer1.html#K5" target="_self">|</a> <font color="Maroon">b</font> <a href="sheffer1.html#R1" target="_self">is_a_complement'_of</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="sheffer1.html#T9" target="_self">Th9</a>, <a class="ref" href="sheffer1.html#T10" target="_self">Th10</a>, <a class="ref" href="sheffer1.html#T11" target="_self">Th11</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:60"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">L</font> is <a href="sheffer1.html#V4" target="_self">complemented'</a>
 
<b>by </b><i/>;<br/>
<a NAME="E7:60"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">L</font> is <a href="lattices.html#V4">join-commutative</a>
 
<b>by </b><i><a class="ref" href="sheffer1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E8:60"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">L</font> is <a href="lattices.html#V6">meet-commutative</a>
 
<b>by </b><i><a class="ref" href="sheffer1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E9:60"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">L</font> is <a href="lattices.html#V11">distributive</a>
 
<b>by </b><i><a class="ref" href="sheffer1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E10:60"/>
<font color="Maroon">L</font> is <a href="sheffer1.html#V3" target="_self">distributive'</a>
 
<b>by </b><i><a class="ref" href="sheffer1.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E11:60"/>
<font color="Maroon">L</font> is  <a href="lattices.html#V17">Boolean</a> <a href="lattices.html#NM1">Lattice</a>
 <b>by </b><i>, <a class="ref" href="sheffer1.html#T8" target="_self">Th8</a>, , <a class="ref" href="sheffer1.html#D9" target="_self">Def9</a>, <a class="ref" href="sheffer1.html#T9" target="_self">Th9</a>, <a class="ref" href="sheffer1.html#T10" target="_self">Th10</a>, </i>;<br/>


</div>
