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<b>let </b><font color="Maroon">S</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel19.html#V2">Lawson</a> <a href="waybel_9.html#NM1">TopLattice</a>, <font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel19.html#V2">Lawson</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="waybel21.html#M1" target="_self">SemilatticeHomomorphism</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:56_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/><a NAME="E2:56_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">X</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> holds  <font color="Maroon">f</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Olive">X</font>
 <b>by </b><i/>;<br/><a NAME="E3:56_1"/>
for <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V2">filtered</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> holds  <font color="Maroon">f</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Olive">X</font>
 <b>by </b><i><a class="ref" href="waybel21.html#T2" target="_self">Th2</a>, </i>;<br/><b>hence </b><a NAME="E4:56_1"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V17">infs-preserving</a>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T71">WAYBEL_0:71</a></i>;<br/><b>thus </b><a NAME="E5:56_1"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <b>by </b><i><a class="ref" href="waybel21.html#T2" target="_self">Th2</a>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:56"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V17">infs-preserving</a>
 ;<br/>

<a NAME="E3:56"/><b>then </b>
for <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> holds  <font color="Maroon">f</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Olive">X</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a></i>;<br/>
<b>hence </b><a NAME="E4:56"/>
(  not <font color="Maroon">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> or <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i/>;<br/>


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