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<b>let </b><font color="Maroon" title="c1">H</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</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">H</font> is <a href="waybel_1.html#V9" title="WAYBEL_1:attr.9">Heyting</a> iff ( <font color="Maroon" title="c1">H</font> is <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a> &amp; <font color="Maroon" title="c1">H</font> is <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> ) )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">H</font> is <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a> &amp; <font color="Maroon" title="c1">H</font> is <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> implies <font color="Maroon" title="c1">H</font> is <a href="waybel_1.html#V9" title="WAYBEL_1:attr.9">Heyting</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:83_1"/><i><font color="Green" title="E55">A1</font></i>: 
<font color="Maroon" title="c1">H</font> is <a href="waybel_1.html#V9" title="WAYBEL_1:attr.9">Heyting</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">H</font> is <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a> &amp; <font color="Maroon" title="c1">H</font> is <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> )</span><br/><a NAME="E2:83_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="c1">H</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#D19" title="WAYBEL_1:def.19">WAYBEL_1:def 19</a></i>;<br/><a NAME="E3:83_1"/><b>then </b><i><font color="Green" title="E56">A2</font></i>: 
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="c1">H</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="c1">H</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="c1">H</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="c1">H</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T67" title="WAYBEL_1:th.67">WAYBEL_1:67</a></i>;<br/><b>thus </b><a NAME="E4:83_1"/>
<font color="Maroon" title="c1">H</font> is <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</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">H</font> is <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a></span><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" title="c1">H</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> &amp; <font color="Maroon" title="c1">H</font> is <a href="waybel_2.html#V1" title="WAYBEL_2:attr.1">satisfying_MC</a> )
 <b>by </b><i><a class="txt" href="waybel_2.html#E3:83_1"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="waybel_2.html#T37" target="_self" title="WAYBEL_2:th.37">Th37</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_2.html#D7" target="_self" title="WAYBEL_2:def.7">WAYBEL_2:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>thus </b><a NAME="E5:83_1"/>
<font color="Maroon" title="c1">H</font> is <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a>
 <b>by </b><i><a class="txt" href="waybel_2.html#E1:83_1"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="waybel_1.html#T69" title="WAYBEL_1:th.69">WAYBEL_1:69</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:83"/><i><font color="Green" title="E55">A3</font></i>: 
( <font color="Maroon" title="c1">H</font> is <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a> &amp; <font color="Maroon" title="c1">H</font> is <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">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">H</font> is <a href="waybel_1.html#V9" title="WAYBEL_1:attr.9">Heyting</a></span><br/>

<b>thus </b><a NAME="E3:83"/>
<font color="Maroon" title="c1">H</font> is   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D19" title="WAYBEL_1:def.19">WAYBEL_1:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">H</font> holds  <font color="Olive">b<sub>1</sub></font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  is <a href="waybel_1.html#V5" title="WAYBEL_1:attr.5">lower_adjoint</a></span><br/>

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">H</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  is <a href="waybel_1.html#V5" title="WAYBEL_1:attr.5">lower_adjoint</a></span><br/>

<a NAME="E4:83"/><i><font color="Green" title="E56">A4</font></i>: 
for <font color="Olive" title="b1">X</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">H</font> holds  <font color="Maroon" title="c2">a</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Olive" title="b1">X</font>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E2:83"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="waybel_2.html#T17" target="_self" title="WAYBEL_2:th.17">Th17</a></i>;<br/>
<a NAME="E5:83"/>
for <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">H</font> holds  <font color="Maroon" title="c2">a</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Olive" title="b1">X</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" title="c3">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">H</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c3">X</font></span><br/>

<a NAME="E1:83_2"/>
<font color="Maroon" title="c2">a</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E2:83"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="waybel_2.html#E63"><i><font color="Green" title="E53">Lm1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:83_2"/>
<font color="Maroon" title="c2">a</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:83"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  is <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E4:83"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="waybel_0.html#T74" title="WAYBEL_0:th.74">WAYBEL_0:74</a></i>;<br/>
<b>hence </b><a NAME="E7:83"/>
<font color="Maroon" title="c2">a</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#T18" title="WAYBEL_1:th.18">WAYBEL_1:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
