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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="lattice3.html#V3">complete</a>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V18">sups-preserving</a>
 ;<br/><a NAME="E2:35_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <span class="p1"><a href="waybel_1.html#K1" target="_self">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></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">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="waybel_1.html#R4" target="_self">is_maximum_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Olive">b<sub>2</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:35"><i><font color="Green">E25</font></i></a>, <a class="ref" href="waybel_1.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E3:35_1"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="orders_3.html#V5">monotone</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="waybel_1.html#NR9" target="_self">has_an_upper_adjoint</a>  )
 <b>by </b><i><a class="ref" href="waybel_1.html#D12" target="_self">Def12</a>, <a class="ref" href="waybel_1.html#T12" target="_self">Th12</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:35"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="orders_3.html#V5">monotone</a>
 ;<br/>

<b>assume </b><a NAME="E4:35"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <span class="p1"><a href="waybel_1.html#K1" target="_self">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></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#D12" target="_self">WAYBEL_1:def 12</a><br/>

<a NAME="E5:35"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_1.html#V5" target="_self">lower_adjoint</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D12" target="_self">Def12</a></i>;<br/>
<b>hence </b><a NAME="E6:35"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V18">sups-preserving</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#T14" target="_self">Th14</a></i>;<br/>


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