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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel19.html#V1" target="_self">lower</a> <a href="waybel_9.html#NM1">TopLattice</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>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E1:18"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<a NAME="E2:18"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V21">filtered-infs-preserving</a>
 
<b>by </b><i><a class="ref" href="waybel19.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E3:18"/><b>then </b>
for <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D36">WAYBEL_0:def 36</a></i>;<br/>
<b>hence </b><a NAME="E4:18"/>
( ex <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st not <font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></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/>


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