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<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c3">d</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c1">S</font>;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c2">T</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon" title="c3">d</font> is <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a>
 ;<br/><a NAME="E2:35_1"/><b>then </b>
 ex <font color="Olive" title="b1">g</font> being  <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> st <br/>( <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Olive" title="b1">g</font>,<font color="Maroon" title="c3">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">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Olive" title="b1">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font> <a href="waybel_1.html#R4" title="WAYBEL_1:pred.4">is_maximum_of</a> <font color="Maroon" title="c3">d</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b2">s</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:35"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="waybel_1.html#T16" target="_self" title="WAYBEL_1:th.16">Th16</a></i>;<br/><b>hence </b><a NAME="E3:35_1"/>
( <font color="Maroon" title="c3">d</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Maroon" title="c3">d</font> <a href="waybel_1.html#NR9" title="WAYBEL_1:NR.9">has_an_upper_adjoint</a>  )
 <b>by </b><i><a class="ref" href="waybel_1.html#D12" target="_self" title="WAYBEL_1:def.12">Def12</a>, <a class="ref" href="waybel_1.html#T12" target="_self" title="WAYBEL_1:th.12">Th12</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:35"/>
<font color="Maroon" title="c3">d</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 ;<br/>

<b>assume </b><a NAME="E4:35"/>
 ex <font color="Olive" title="b1">g</font> being  <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> st <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Olive" title="b1">g</font>,<font color="Maroon" title="c3">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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D12" target="_self" title="WAYBEL_1:def.12">WAYBEL_1:def 12</a><br/>

<a NAME="E5:35"/><b>then </b>
<font color="Maroon" title="c3">d</font> is <a href="waybel_1.html#V5" title="WAYBEL_1:attr.5">lower_adjoint</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D12" target="_self" title="WAYBEL_1:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E6:35"/>
<font color="Maroon" title="c3">d</font> is <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#T14" target="_self" title="WAYBEL_1:th.14">Th14</a></i>;<br/>


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