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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:72_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_2.html#V2" target="_self">meet-continuous</a>
 ;<br/><a NAME="E2:72_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<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="ref" href="waybel_2.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E3:72_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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 <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="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>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="yellow_4.html#K4">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <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="E1:72_1_1"/>
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><br/>for <font color="Olive">b<sub>2</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> st <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E2:72_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="waybel_2.html#T45" target="_self">Th45</a></i>;<br/>
<a NAME="E2:72_1_1"/><b>then </b>
 <a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i><a class="ref" href="waybel_2.html#T46" target="_self">Th46</a></i>;<br/>
<b>hence </b><a NAME="E3:72_1_1"/>
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_2.html#T43" target="_self">Th43</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:72_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> holds <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="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>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="yellow_4.html#K4">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 ;<br/><b>hence </b><a NAME="E5:72_1"/>
(  <a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V19">meet-preserving</a> &amp;  <a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V20">join-preserving</a> )
 <b>by </b><i><a class="ref" href="waybel_2.html#T39" target="_self">Th39</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:72"/><i><font color="Green">E49</font></i>: 
(  <a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V19">meet-preserving</a> &amp;  <a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V20">join-preserving</a> )
 ;<br/>

<b>thus </b><a NAME="E3:72"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V24">up-complete</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_2.html#D7" target="_self">WAYBEL_2:def 7</a><br/>

<a NAME="E4:72"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> holds <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="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>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="yellow_4.html#K4">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E2:72"><i><font color="Green">E49</font></i></a>, <a class="ref" href="waybel_2.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E5:72"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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 <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="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>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="yellow_4.html#K4">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_2.html#T40" target="_self">Th40</a></i>;<br/>
<b>hence </b><a NAME="E6:72"/>
<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="ref" href="waybel_2.html#T44" target="_self">Th44</a></i>;<br/>


</div>
