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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:83_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V9">Heyting</a>
 ;<br/><a NAME="E2:83_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></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#D19">WAYBEL_1:def 19</a></i>;<br/><a NAME="E3:83_1"/><b>then </b><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></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">b<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>3</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T67">WAYBEL_1:67</a></i>;<br/><b>thus </b><a NAME="E4:83_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_2.html#V2" target="_self">meet-continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:83_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V24">up-complete</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_2.html#V1" target="_self">satisfying_MC</a> )
 <b>by </b><i><a class="txt" href="waybel_2.html#E3:83_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="waybel_2.html#T37" target="_self">Th37</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_2.html#D7" target="_self">WAYBEL_2:def 7</a><br/>


</div><b>end;</b></div><b>thus </b><a NAME="E5:83_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V2">distributive</a>
 <b>by </b><i><a class="txt" href="waybel_2.html#E1:83_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="waybel_1.html#T69">WAYBEL_1:69</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:83"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_2.html#V2" target="_self">meet-continuous</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V2">distributive</a> )
 ;<br/>

<b>thus </b><a NAME="E3:83"/>
<font color="Maroon">c<sub>1</sub></font> is   <a href="waybel_0.html#NM8">LATTICE</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D19">WAYBEL_1:def 19</a><br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E4:83"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="waybel_1.html#K4">"/\"</a>  <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E2:83"><i><font color="Green">E55</font></i></a>, <a class="ref" href="waybel_2.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E5:83"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="waybel_1.html#K4">"/\"</a>  <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:83_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_1.html#K4">"/\"</a>  is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E2:83"><i><font color="Green">E55</font></i></a>, <a class="txt" href="waybel_2.html#E63"><i><font color="Green">Lemma53</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:83_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_1.html#K4">"/\"</a>  <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:83"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_1.html#K4">"/\"</a>  is <a href="waybel_0.html#V18">sups-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E4:83"><i><font color="Green">E56</font></i></a>, <a class="ref" href="waybel_0.html#T74">WAYBEL_0:74</a></i>;<br/>
<b>hence </b><a NAME="E7:83"/>
<font color="Maroon">c<sub>2</sub></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#T18">WAYBEL_1:18</a></i>;<br/>


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