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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:74_1"/>
<font color="Maroon" title="c1">L</font> is <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a>
 ;<br/><a NAME="E2:74_1"/><b>then </b>
(  <a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V19" title="WAYBEL_0:attr.19">meet-preserving</a> &amp;  <a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V20" title="WAYBEL_0:attr.20">join-preserving</a> )
 <b>by </b><i><a class="ref" href="waybel_2.html#T49" target="_self" title="WAYBEL_2:th.49">Th49</a></i>;<br/><b>hence </b><a NAME="E3:74_1"/>
for <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being  <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">I1</font></span>)</span> <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="b2">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">I1</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Olive" title="b2">I2</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_2.html#T38" target="_self" title="WAYBEL_2:th.38">Th38</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:74"/>
for <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being  <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">I1</font></span>)</span> <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="b2">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">I1</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Olive" title="b2">I2</font></span>)</span>
 ;<br/>

<a NAME="E3:74"/><b>then </b>
for <font color="Olive" title="b1">D1</font>, <font color="Olive" title="b2">D2</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">L</font> holds  <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">D1</font></span>)</span> <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="b2">D2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">D1</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Olive" title="b2">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_2.html#T40" target="_self" title="WAYBEL_2:th.40">Th40</a></i>;<br/>
<b>hence </b><a NAME="E4:74"/>
( <font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="waybel_2.html#V1" title="WAYBEL_2:attr.1">satisfying_MC</a> )
 <b>by </b><i><a class="ref" href="waybel_2.html#T44" target="_self" title="WAYBEL_2:th.44">Th44</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><br/>


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