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<b>let </b><font color="Maroon">L</font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:54_1_1"/>
<font color="Maroon">L</font> is <a href="waybel_5.html#V1" target="_self">completely-distributive</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">L</font> = <font color="Maroon">L</font> as  <a href="waybel_5.html#V1" target="_self">completely-distributive</a> <a href="waybel_0.html#NM8">LATTICE</a> ;<br/>
<a NAME="E3:54_1_1"/>
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><br/> for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:54_1_1"/><b>then </b>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">x</font> <a href="waybel_1.html#K4">"/\"</a>  <a href="waybel_1.html#NR9" target="_self">has_an_upper_adjoint</a> 
 
<b>by </b><i><a class="ref" href="waybel_1.html#T67">WAYBEL_1:67</a></i>;<br/>
<b>hence </b><a NAME="E5:54_1_1"/>
<font color="Maroon">L</font> is <a href="waybel_1.html#V9">Heyting</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#D19">WAYBEL_1:def 19</a></i>;<br/>


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