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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM6" title="WAYBEL_0:NM.6">Semilattice</a>;<br/>

<b>set </b><font color="Maroon" title="c2">r</font> =  <a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font>;<br/>
<b>assume </b><a NAME="E1:7"/>
(  <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#V17" title="WAYBEL_0:attr.17">infs-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#V18" title="WAYBEL_0:attr.18">sups-preserving</a> )
 ;<br/>

<a NAME="E2:7"/><b>then </b>
 ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> st <br/>( <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b1">d</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a> &amp; ( for <font color="Olive" title="b2">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b1">d</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b2">t</font> <a href="waybel_1.html#R3" title="WAYBEL_1:pred.3">is_minimum_of</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b2">t</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="waybel_1.html#T15" title="WAYBEL_1:th.15">WAYBEL_1:15</a></i>;<br/>
<b>hence </b><a NAME="E3:7"/>
 <a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">L</font> is <a href="waybel_1.html#V4" title="WAYBEL_1:attr.4">upper_adjoint</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#D11" title="WAYBEL_1:def.11">WAYBEL_1:def 11</a></i>;<br/>


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