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

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="sheffer1.html#V9" title="SHEFFER1:attr.9">properly_defined</a> <a href="sheffer1.html#V10" title="SHEFFER1:attr.10">satisfying_Sheffer_1</a> <a href="sheffer1.html#V11" title="SHEFFER1:attr.11">satisfying_Sheffer_2</a> <a href="sheffer1.html#V12" title="SHEFFER1:attr.12">satisfying_Sheffer_3</a>  <a href="sheffer1.html#L3" title="SHEFFER1:struct.3">ShefferOrthoLattStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a></span><br/>

<a NAME="E1:60"/>
 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="c1">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="c1">L</font> holds <br/> ( <font color="Olive" title="b1">c</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</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#K1" title="LATTICES:func.1">"\/"</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="60_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>
<b>take </b><font color="Maroon" title="c3">c</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c1">L</font> holds <br/> ( <font color="Maroon" title="c3">c</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</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#K1" title="LATTICES:func.1">"\/"</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/>

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">c</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c4">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )</span><br/>

<b>thus </b><a NAME="E2:60_1"/>
<font color="Maroon" title="c3">c</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>proof </b></a><div class="add">

<font color="Maroon" title="c3">c</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#T33" target="_self" title="SHEFFER1:th.33">Th33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D13" target="_self" title="SHEFFER1:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#T32" target="_self" title="SHEFFER1:th.32">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D14" target="_self" title="SHEFFER1:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="sheffer1.html#D13" target="_self" title="SHEFFER1:def.13">Def13</a></i>
;<br/>
<b>hence </b><a NAME="E2:60_1_1"/>
<font color="Maroon" title="c3">c</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E3:60_1"/>
<font color="Maroon" title="c4">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2"><b>proof </b></a><div class="add">

<font color="Maroon" title="c4">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#T33" target="_self" title="SHEFFER1:th.33">Th33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D13" target="_self" title="SHEFFER1:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D14" target="_self" title="SHEFFER1:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="sheffer1.html#D13" target="_self" title="SHEFFER1:def.13">Def13</a></i>
;<br/>
<b>hence </b><a NAME="E2:60_1_2"/>
<font color="Maroon" title="c4">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div><b>end;</b></div>
<a NAME="E2:60"/><b>then </b><i><font color="Green" title="E44">A1</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="E3:60"/>
 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="c1">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="c1">L</font> holds <br/> ( <font color="Olive" title="b1">c</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</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#K2" title="LATTICES:func.2">"/\"</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="60_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>
<b>take </b><font color="Maroon" title="c3">c</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c1">L</font> holds <br/> ( <font color="Maroon" title="c3">c</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</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#K2" title="LATTICES:func.2">"/\"</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/>

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">c</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )</span><br/>

<b>thus </b><a NAME="E2:60_2"/>
<font color="Maroon" title="c3">c</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font></span><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" title="c5">X</font> = <font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> ;<br/>
<font color="Maroon" title="c3">c</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> 
<b>by </b><i><a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> 
<b>by </b><i><a class="ref" href="sheffer1.html#T32" target="_self" title="SHEFFER1:th.32">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> 
<b>by </b><i><a class="ref" href="sheffer1.html#D14" target="_self" title="SHEFFER1:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="sheffer1.html#D13" target="_self" title="SHEFFER1:def.13">Def13</a></i>
;<br/>
<b>hence </b><a NAME="E2:60_2_1"/>
<font color="Maroon" title="c3">c</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E3:60_2"/>
<font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c5">X</font> = <font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> ;<br/>
<font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> 
<b>by </b><i><a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> 
<b>by </b><i><a class="ref" href="sheffer1.html#D14" target="_self" title="SHEFFER1:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="sheffer1.html#D13" target="_self" title="SHEFFER1:def.13">Def13</a></i>
;<br/>
<b>hence </b><a NAME="E2:60_2_2"/>
<font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div><b>end;</b></div>
<a NAME="E4:60"/><b>then </b><i><font color="Green" title="E45">A2</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/>
<a NAME="E5:60"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  ex <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="c1">L</font> st <font color="Olive" title="b2">a</font> <a href="sheffer1.html#R1" title="SHEFFER1:pred.1">is_a_complement'_of</a> <font color="Olive" title="b1">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" title="c2">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <font color="Olive" title="b1">a</font> <a href="sheffer1.html#R1" title="SHEFFER1:pred.1">is_a_complement'_of</a> <font color="Maroon" title="c2">b</font></span><br/>

<a NAME="E1:60_3"/><i><font color="Green" title="E46">A3</font></i>: 
 <a href="sheffer1.html#K1" title="SHEFFER1:func.1">Top'</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">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" title="c3">X</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:60_3_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="c1">L</font> holds <br/> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</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#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )</span><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" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font></span><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" title="c5">Y</font> = <font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> ;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> 
<b>by </b><i><a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> 
<b>by </b><i><a class="ref" href="sheffer1.html#T32" target="_self" title="SHEFFER1:th.32">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> 
<b>by </b><i><a class="ref" href="sheffer1.html#D14" target="_self" title="SHEFFER1:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="sheffer1.html#D13" target="_self" title="SHEFFER1:def.13">Def13</a></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" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:60_3_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c5">Y</font> = <font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> ;<br/>
<font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> 
<b>by </b><i><a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> 
<b>by </b><i><a class="ref" href="sheffer1.html#D14" target="_self" title="SHEFFER1:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="sheffer1.html#D13" target="_self" title="SHEFFER1:def.13">Def13</a></i>
;<br/>
<b>hence </b><a NAME="E2:60_3_1_1_2"/>
<font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E3:60_3_1_1"/>
verum
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:60_3_1"/>
 <a href="sheffer1.html#K1" title="SHEFFER1:func.1">Top'</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sheffer1.html#E4:60"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="sheffer1.html#D2" target="_self" title="SHEFFER1:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:60_3"/><i><font color="Green" title="E47">A4</font></i>: 
 <a href="sheffer1.html#K2" title="SHEFFER1:func.2">Bot'</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">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" title="c3">X</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:60_3_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="c1">L</font> holds <br/> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</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#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c4">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )</span><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" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font></span><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" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#T33" target="_self" title="SHEFFER1:th.33">Th33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D13" target="_self" title="SHEFFER1:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#T32" target="_self" title="SHEFFER1:th.32">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D14" target="_self" title="SHEFFER1:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="sheffer1.html#D13" target="_self" title="SHEFFER1:def.13">Def13</a></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" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:60_3_2_1"/>
<font color="Maroon" title="c4">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c4">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span> <a href="sheffer1.html#K6" title="SHEFFER1:func.6">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#T33" target="_self" title="SHEFFER1:th.33">Th33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D13" target="_self" title="SHEFFER1:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D14" target="_self" title="SHEFFER1:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="ref" href="sheffer1.html#D13" target="_self" title="SHEFFER1:def.13">Def13</a></i>
;<br/>
<b>hence </b><a NAME="E2:60_3_2_1_2"/>
<font color="Maroon" title="c4">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E3:60_3_2_1"/>
verum
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:60_3_2"/>
 <a href="sheffer1.html#K2" title="SHEFFER1:func.2">Bot'</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sheffer1.html#E2:60"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="sheffer1.html#D4" target="_self" title="SHEFFER1:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font>;<br/>
<b>take </b>
<font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font> <a href="sheffer1.html#R1" title="SHEFFER1:pred.1">is_a_complement'_of</a> <font color="Maroon" title="c2">b</font></span><br/>

<i><font color="Green" title="E48">A5</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">b</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>
<br/>.= 
 <a href="sheffer1.html#K1" title="SHEFFER1:func.1">Top'</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="txt" href="sheffer1.html#E1:60_3"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="sheffer1.html#T32" target="_self" title="SHEFFER1:th.32">Th32</a></i>
;<br/>
<a NAME="E4:60_3"/><i><font color="Green" title="E49">A6</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sheffer1.html#K1" title="SHEFFER1:func.1">Top'</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="sheffer1.html#E1:60_3"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>;<br/>
<i><font color="Green" title="E50">A7</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c2">b</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span>
<b>by </b><i><a class="ref" href="sheffer1.html#T32" target="_self" title="SHEFFER1:th.32">Th32</a></i>
<br/>.= 
 <a href="sheffer1.html#K2" title="SHEFFER1:func.2">Bot'</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="txt" href="sheffer1.html#E2:60_3"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="sheffer1.html#T32" target="_self" title="SHEFFER1:th.32">Th32</a></i>
;<br/>
<a NAME="E6:60_3"/>
<font color="Maroon" title="c2">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sheffer1.html#K2" title="SHEFFER1:func.2">Bot'</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="sheffer1.html#E2:60_3"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="sheffer1.html#D12" target="_self" title="SHEFFER1:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E7:60_3"/>
<font color="Maroon" title="c2">b</font> <a href="sheffer1.html#K5" title="SHEFFER1:func.5">|</a> <font color="Maroon" title="c2">b</font> <a href="sheffer1.html#R1" title="SHEFFER1:pred.1">is_a_complement'_of</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="sheffer1.html#E3:60_3"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="sheffer1.html#E4:60_3"><i><font color="Green" title="E49">A6</font></i></a>, <a class="txt" href="sheffer1.html#E5:60_3"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="sheffer1.html#D6" target="_self" title="SHEFFER1:def.6">Def6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:60"/><b>then </b><i><font color="Green" title="E46">A8</font></i>: 
<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="ref" href="sheffer1.html#D7" target="_self" title="SHEFFER1:def.7">Def7</a></i>;<br/>
<a NAME="E7:60"/><i><font color="Green" title="E47">A9</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a>
 
<b>by </b><i><a class="ref" href="sheffer1.html#T34" target="_self" title="SHEFFER1:th.34">Th34</a></i>;<br/>
<a NAME="E8:60"/><i><font color="Green" title="E48">A10</font></i>: 
<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="ref" href="sheffer1.html#T35" target="_self" title="SHEFFER1:th.35">Th35</a></i>;<br/>
<a NAME="E9:60"/><i><font color="Green" title="E49">A11</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a>
 
<b>by </b><i><a class="ref" href="sheffer1.html#T36" target="_self" title="SHEFFER1:th.36">Th36</a></i>;<br/>
<a NAME="E10:60"/>
<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#T37" target="_self" title="SHEFFER1:th.37">Th37</a></i>;<br/>
<b>hence </b><a NAME="E11:60"/>
<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#E2:60"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="sheffer1.html#E4:60"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="sheffer1.html#E6:60"><i><font color="Green" title="E46">A8</font></i></a>, <a class="txt" href="sheffer1.html#E7:60"><i><font color="Green" title="E47">A9</font></i></a>, <a class="txt" href="sheffer1.html#E8:60"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="sheffer1.html#E9:60"><i><font color="Green" title="E49">A11</font></i></a>, <a class="ref" href="sheffer1.html#T26" target="_self" title="SHEFFER1:th.26">Th26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
