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<b>let </b><font color="Maroon">S</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:112"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">T</font> is <a href="waybel_8.html#V2">algebraic</a>
 ;<br/>

<a NAME="E2:112"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">T</font> is <a href="waybel_3.html#V3">continuous</a>
 
<b>by </b><i>, <a class="ref" href="waybel_8.html#T7">WAYBEL_8:7</a></i>;<br/>
<b>let </b><font color="Maroon">g</font> be  <a href="waybel_0.html#V17">infs-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:112_1"/>
<font color="Maroon">g</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 ;<br/><a NAME="E2:112_1"/><b>then </b>
 <a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font> is <a href="waybel34.html#V1" target="_self">waybelow-preserving</a>
 <b>by </b><i/>;<br/><a NAME="E3:112_1"/><b>then </b>
 <a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font> is <a href="waybel34.html#V3" target="_self">compact-preserving</a>
 <b>by </b><i><a class="ref" href="waybel34.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E4:112_1"/>
<span class="p1">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">T</font></span>)</span> is  <a href="waybel34.html#V4" target="_self">finite-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:112"/>
<span class="p1">(<span class="default"><a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">T</font></span>)</span> is  <a href="waybel34.html#V4" target="_self">finite-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">S</font></span>)</span>
 ;<br/>

<a NAME="E5:112"/><b>then </b>
 <a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font> is <a href="waybel34.html#V3" target="_self">compact-preserving</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:112"/><b>then </b>
 <a href="waybel34.html#K1" target="_self">LowerAdj</a> <font color="Maroon">g</font> is <a href="waybel34.html#V1" target="_self">waybelow-preserving</a>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:112"/>
<font color="Maroon">g</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <b>by </b><i>, </i>;<br/>


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