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<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</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="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</a> implies <font color="Maroon" title="c1">L</font> is <a href="waybel_1.html#V9" title="WAYBEL_1:attr.9">Heyting</a> )</span><br/>

<b>assume </b><a NAME="E1:54_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</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="waybel_1.html#V9" title="WAYBEL_1:attr.9">Heyting</a></span><br/>

<b>then reconsider </b><font color="Maroon" title="c2">L</font> = <font color="Maroon" title="c1">L</font> as  <a href="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> ;<br/>
<a NAME="E3:54_1_1"/>
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> holds  <font color="Olive" title="b2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">y</font></span>)</span> where <font color="Olive" title="b3">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> : <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> </span>}</span> ,<font color="Maroon" title="c2">L</font>
 
<b>by </b><i><a class="ref" href="waybel_5.html#T26" target="_self" title="WAYBEL_5:th.26">Th26</a></i>;<br/>
<a NAME="E4:54_1_1"/><b>then </b>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> holds  <font color="Olive" title="b1">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  <a href="waybel_1.html#NR9" title="WAYBEL_1:NR.9">has_an_upper_adjoint</a> 
 
<b>by </b><i><a class="ref" href="waybel_1.html#T67" title="WAYBEL_1:th.67">WAYBEL_1:67</a></i>;<br/>
<b>hence </b><a NAME="E5:54_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="waybel_1.html#V9" title="WAYBEL_1:attr.9">Heyting</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#D19" title="WAYBEL_1:def.19">WAYBEL_1:def 19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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