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<b>let </b><font color="Maroon">L</font> be  <a href="waybel_1.html#V11">Boolean</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:30"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">a</font> is <a href="waybel15.html#V1" target="_self">atom</a>
 ;<br/>

<b>thus </b><a NAME="E2:30"/>
( <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> implies  not <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:30_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E2:30_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font>
 ;<br/>

<font color="Maroon">a</font> = 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_5.html#T37">YELLOW_5:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#T6">WAYBEL_1:6</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
<br/>.= 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="yellow_5.html#T37">YELLOW_5:37</a></i>
;<br/>
<b>hence </b><a NAME="E4:30_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E3:30"/>
( not <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font> implies <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:30_2"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E2:30_2"/><i><font color="Green">E47</font></i>: 
not <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E3:30_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">a</font> is <a href="waybel_6.html#V6">co-prime</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:30_2"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/>
<a NAME="E5:30_2"/><b>then </b>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_5.html#T37">YELLOW_5:37</a></i>;<br/>
<b>hence </b><a NAME="E6:30_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="waybel14.html#T14">WAYBEL14:14</a></i>;<br/>


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


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