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<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <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>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> holds <br/>( ( <font color="Olive" title="b2">b</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b3">c</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Olive" title="b1">a</font> implies <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">b</font> ) &amp; ( <font color="Olive" title="b2">b</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b3">c</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Olive" title="b1">a</font> implies <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">b</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">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"> for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">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 <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> holds <br/>( ( <font color="Olive" title="b1">b</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b2">c</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">c</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">b</font> ) &amp; ( <font color="Olive" title="b1">b</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b2">c</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">c</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">b</font> ) )</span><br/><b>let </b><font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</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">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> implies ( ( <font color="Maroon" title="c3">b</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font> ) &amp; ( <font color="Maroon" title="c3">b</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font> ) ) )</span><br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</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">b</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font> ) &amp; ( <font color="Maroon" title="c3">b</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font> ) )</span><br/>

<div><i><font color="Green" title="E13">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_1"/><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">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="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font></span><br/><a NAME="E2:20_1"/><b>then </b><i><font color="Green" title="E15">A4</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">c</font> )
 <b>by </b><i><a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</a></i>;<br/><font color="Maroon" title="c2">a</font> = 
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="lattices.html#T18" title="LATTICES:th.18">LATTICES:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="lattice6.html#E2:20_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice6.html#E2:20_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">c</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">c</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#T18" title="LATTICES:th.18">LATTICES:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a></i>
;<br/><a NAME="E4:20_1"/><b>then </b><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>;<br/><a NAME="E5:20_1"/><i><font color="Green" title="E17">A6</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="ref" href="lattices.html#T23" title="LATTICES:th.23">LATTICES:23</a></i>;<br/><a NAME="E6:20_1"/>
not <font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_2"/>
<font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:20_1_2"/><b>then </b>
<font color="Maroon" title="c4">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>;<br/>
<a NAME="E3:20_1_2"/><b>then </b>
( <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:20_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="lattice6.html#E2:20_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E4:20_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice6.html#E1:20"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="lattice6.html#E1:20_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:20_1"/>
<font color="Maroon" title="c3">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E1:20_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="lattice6.html#E4:20_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="lattice6.html#E5:20_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_2"/><i><font color="Green" title="E14">A7</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">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="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font></span><br/><a NAME="E2:20_2"/><b>then </b><i><font color="Green" title="E15">A8</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</a></i>;<br/><font color="Maroon" title="c2">a</font> = 
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="lattices.html#T17" title="LATTICES:th.17">LATTICES:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="lattice6.html#E2:20_2"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">a</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice6.html#E2:20_2"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">c</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">c</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#T17" title="LATTICES:th.17">LATTICES:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a></i>
;<br/><a NAME="E4:20_2"/><b>then </b><i><font color="Green" title="E16">A9</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></i>;<br/><a NAME="E5:20_2"/><i><font color="Green" title="E17">A10</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a></i>;<br/><a NAME="E6:20_2"/>
not <font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2_2"/>
<font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:20_2_2"/><b>then </b>
<font color="Maroon" title="c3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></i>;<br/>
<a NAME="E3:20_2_2"/><b>then </b>
( <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:20_2"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="lattice6.html#E2:20_2"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E4:20_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice6.html#E1:20"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="lattice6.html#E1:20_2"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:20_2"/>
<font color="Maroon" title="c3">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E1:20_2"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="lattice6.html#E4:20_2"><i><font color="Green" title="E16">A9</font></i></a>, <a class="txt" href="lattice6.html#E5:20_2"><i><font color="Green" title="E17">A10</font></i></a>, <a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:20"/>
( ( <font color="Maroon" title="c3">b</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">b</font> ) &amp; ( <font color="Maroon" title="c3">b</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font> ) )
 <b>by </b><i><a class="txt" href="lattice6.html#E2:20"><i><font color="Green" title="E13">A2</font></i></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>
