<?xml version="1.0"?>
<div class="add">

<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <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 <font color="Maroon" title="c1">S</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> holds <br/>( <font color="Olive" title="b1">g</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> iff ( <font color="Olive" title="b1">g</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Olive" title="b1">g</font> <a href="waybel_1.html#NR8" title="WAYBEL_1:NR.8">has_a_lower_adjoint</a>  ) )</span><br/><b>let </b><font color="Maroon" title="c3">g</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">S</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> implies ( <font color="Maroon" title="c3">g</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> iff ( <font color="Maroon" title="c3">g</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Maroon" title="c3">g</font> <a href="waybel_1.html#NR8" title="WAYBEL_1:NR.8">has_a_lower_adjoint</a>  ) ) )</span><br/>



<b>assume </b><a NAME="E1:34"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c1">S</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">g</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> iff ( <font color="Maroon" title="c3">g</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Maroon" title="c3">g</font> <a href="waybel_1.html#NR8" title="WAYBEL_1:NR.8">has_a_lower_adjoint</a>  ) )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">g</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Maroon" title="c3">g</font> <a href="waybel_1.html#NR8" title="WAYBEL_1:NR.8">has_a_lower_adjoint</a>  implies <font color="Maroon" title="c3">g</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon" title="c3">g</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">g</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Maroon" title="c3">g</font> <a href="waybel_1.html#NR8" title="WAYBEL_1:NR.8">has_a_lower_adjoint</a>  )</span><br/><a NAME="E2:34_1"/><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="c2">T</font>,<font color="Maroon" title="c1">S</font> st <br/>( <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">g</font>,<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="c2">T</font> holds  <font color="Olive" title="b1">d</font> <a href="funct_2.html#K3" title="FUNCT_2: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> <font color="Maroon" title="c3">g</font> <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="txt" href="waybel_1.html#E1:34"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="waybel_1.html#T15" target="_self" title="WAYBEL_1:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E3:34_1"/>
( <font color="Maroon" title="c3">g</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; <font color="Maroon" title="c3">g</font> <a href="waybel_1.html#NR8" title="WAYBEL_1:NR.8">has_a_lower_adjoint</a>  )
 <b>by </b><i><a class="ref" href="waybel_1.html#D11" target="_self" title="WAYBEL_1:def.11">Def11</a>, <a class="ref" href="waybel_1.html#T11" target="_self" title="WAYBEL_1:th.11">Th11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:34"/>
<font color="Maroon" title="c3">g</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">g</font> <a href="waybel_1.html#NR8" title="WAYBEL_1:NR.8">has_a_lower_adjoint</a>  or <font color="Maroon" title="c3">g</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> )</span><br/>

<b>assume </b><a NAME="E4:34"/>
 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="c2">T</font>,<font color="Maroon" title="c1">S</font> st <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">g</font>,<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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D11" target="_self" title="WAYBEL_1:def.11">WAYBEL_1:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">g</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a></span><br/>

<a NAME="E5:34"/><b>then </b>
<font color="Maroon" title="c3">g</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" target="_self" title="WAYBEL_1:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E6:34"/>
<font color="Maroon" title="c3">g</font> is <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#T13" target="_self" title="WAYBEL_1:th.13">Th13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
