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<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel19.html#V1" title="WAYBEL19:attr.1">lower</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>;<br/><b>let </b><font color="Maroon" title="c3">f</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>;<br/>



<b>assume </b><a NAME="E1:18"/>
<font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ;<br/>

<a NAME="E2:18"/><b>then </b>
<font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V21" title="WAYBEL_0:attr.21">filtered-infs-preserving</a>
 
<b>by </b><i><a class="ref" href="waybel19.html#T11" target="_self" title="WAYBEL19:th.11">Th11</a></i>;<br/>
<a NAME="E3:18"/><b>then </b>
for <font color="Olive" title="b1">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c3">f</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Olive" title="b1">F</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D36" title="WAYBEL_0:def.36">WAYBEL_0:def 36</a></i>;<br/>
<b>hence </b><a NAME="E4:18"/>
(  ex <font color="Olive" title="b1">X</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> st not <font color="Maroon" title="c3">f</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <font color="Olive" title="b1">X</font> or <font color="Maroon" title="c3">f</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_0.html#T71" title="WAYBEL_0:th.71">WAYBEL_0:71</a></i>;<br/>


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