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<b>let </b><font color="Maroon">L</font> be  <a href="lattices.html#V13">lower-bounded</a> <a href="lattice6.html#V2" target="_self">co-noetherian</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  holds not <font color="Olive">b</font> <a href="lattice6.html#NR2" target="_self">is-lower-neighbour-of</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E2:26_1"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Olive">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:26_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font>
 ;<br/>
<a NAME="E3:26_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">c</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="lattices.html#T23">LATTICES:23</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font>
 ;<br/><div class="add"><a NAME="E2:26_1_1_1_1"/><b>then </b>
 ex <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">c</font> <a href="lattices.html#R3">[=</a> <font color="Olive">d</font> &amp; <font color="Olive">d</font> <a href="lattice6.html#NR2" target="_self">is-lower-neighbour-of</a> <font color="Maroon">a</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:26_1_1_1_1"/>
( <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:26_1_1_1_2"/>
( <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E3:26_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="lattices.html#D16">LATTICES:def 16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:26"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> iff for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  holds not <font color="Olive">b</font> <a href="lattice6.html#NR2" target="_self">is-lower-neighbour-of</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/>


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