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<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:112"/><i><font color="Green" title="E68">A1</font></i>: 
<font color="Maroon" title="c2">T</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 ;<br/>

<a NAME="E2:112"/><i><font color="Green" title="E69">A2</font></i>: 
<font color="Maroon" title="c2">T</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel34.html#E1:112"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="waybel_8.html#T7" title="WAYBEL_8:th.7">WAYBEL_8:7</a></i>;<br/>
<b>let </b><font color="Maroon" title="c3">g</font> be  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:112_1"/>
<font color="Maroon" title="c3">g</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 ;<br/><a NAME="E2:112_1"/><b>then </b>
 <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font> is <a href="waybel34.html#V1" title="WAYBEL34:attr.1">waybelow-preserving</a>
 <b>by </b><i><a class="ref" href="waybel34.html#T24" target="_self" title="WAYBEL34:th.24">Th24</a></i>;<br/><a NAME="E3:112_1"/><b>then </b>
 <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font> is <a href="waybel34.html#V3" title="WAYBEL34:attr.3">compact-preserving</a>
 <b>by </b><i><a class="ref" href="waybel34.html#T60" target="_self" title="WAYBEL34:th.60">Th60</a></i>;<br/><b>hence </b><a NAME="E4:112_1"/>
<span class="p1">(<span class="default"><a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c2">T</font></span>)</span> is  <a href="waybel34.html#V4" title="WAYBEL34:attr.4">finite-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c2">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel34.html#T70" target="_self" title="WAYBEL34:th.70">Th70</a></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" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c2">T</font></span>)</span> is  <a href="waybel34.html#V4" title="WAYBEL34:attr.4">finite-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c2">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">S</font></span>)</span>
 ;<br/>

<a NAME="E5:112"/><b>then </b>
 <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font> is <a href="waybel34.html#V3" title="WAYBEL34:attr.3">compact-preserving</a>
 
<b>by </b><i><a class="ref" href="waybel34.html#T70" target="_self" title="WAYBEL34:th.70">Th70</a></i>;<br/>
<a NAME="E6:112"/><b>then </b>
 <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <font color="Maroon" title="c3">g</font> is <a href="waybel34.html#V1" title="WAYBEL34:attr.1">waybelow-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel34.html#E1:112"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="waybel34.html#T61" target="_self" title="WAYBEL34:th.61">Th61</a></i>;<br/>
<b>hence </b><a NAME="E7:112"/>
<font color="Maroon" title="c3">g</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 <b>by </b><i><a class="txt" href="waybel34.html#E2:112"><i><font color="Green" title="E69">A2</font></i></a>, <a class="ref" href="waybel34.html#T25" target="_self" title="WAYBEL34:th.25">Th25</a></i>;<br/>


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