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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>, <font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:34"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">S</font> is <a href="lattice3.html#V3">complete</a>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon">g</font> is <a href="waybel_0.html#V17">infs-preserving</a>
 ;<br/><a NAME="E2:34_1"/><b>then </b>
 ex <font color="Olive">d</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font> st <br/>( <span class="p1"><a href="waybel_1.html#K1" target="_self">[</a><span class="default"><font color="Maroon">g</font>,<font color="Olive">d</font></span><a href="waybel_1.html#K1" target="_self">]</a></span> is <a href="waybel_1.html#V3" target="_self">Galois</a> &amp; ( for <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> holds  <font color="Olive">d</font> <a href="funct_2.html#K3">.</a> <font color="Olive">t</font> <a href="waybel_1.html#R3" target="_self">is_minimum_of</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">t</font></span>)</span> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:34_1"/>
( <font color="Maroon">g</font> is <a href="orders_3.html#V5">monotone</a> &amp; <font color="Maroon">g</font> <a href="waybel_1.html#NR8" target="_self">has_a_lower_adjoint</a>  )
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:34"/>
<font color="Maroon">g</font> is <a href="orders_3.html#V5">monotone</a>
 ;<br/>

<b>assume </b><a NAME="E4:34"/>
 ex <font color="Olive">d</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font> st <span class="p1"><a href="waybel_1.html#K1" target="_self">[</a><span class="default"><font color="Maroon">g</font>,<font color="Olive">d</font></span><a href="waybel_1.html#K1" target="_self">]</a></span> is <a href="waybel_1.html#V3" target="_self">Galois</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D11" target="_self">WAYBEL_1:def 11</a><br/>

<a NAME="E5:34"/><b>then </b>
<font color="Maroon">g</font> is <a href="waybel_1.html#V4" target="_self">upper_adjoint</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:34"/>
<font color="Maroon">g</font> is <a href="waybel_0.html#V17">infs-preserving</a>
 <b>by </b><i/>;<br/>


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