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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="waybel_1.html#V8">kernel</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:55"/><i><font color="Green">E37</font></i>: 
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 <b>and </b><br/><a NAME="E2:55"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <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><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">b<sub>2</sub></font> iff <font color="Olive">b<sub>3</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">b<sub>4</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E3:55"/><i><font color="Green">E39</font></i>: 
 <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V17">infs-preserving</a>
 
<b>by </b><i><a class="ref" href="waybel34.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E4:55"/>
 <a href="waybel34.html#K1" target="_self">LowerAdj</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="waybel34.html#V1" target="_self">waybelow-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel34.html#D8" target="_self">WAYBEL34:def 8</a><br/>

<a NAME="E1:55_1"/>
 <a href="waybel34.html#K1" target="_self">LowerAdj</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="waybel34.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E2:55_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>;<br/>
<b>hence </b><a NAME="E3:55_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>5</sub></font> implies <span class="p1">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <span class="p1">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="waybel34.html#E2:55"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:55"/><b>then </b>
 <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel34.html#E1:55"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel34.html#E3:55"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel34.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E6:55"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <b>by </b><i><a class="ref" href="waybel34.html#T34" target="_self">Th34</a></i>;<br/>


</div>
