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<b>let </b><font color="Maroon">S</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel11.html#V4">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a>, <font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel11.html#V4">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>



<b>thus </b><a NAME="E1:54"/>
( <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> implies <font color="Maroon">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<a NAME="E2:54_1"/><b>then </b>
for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">S</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <font color="Olive">N</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Olive">N</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:54_1"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:54"/>
( <font color="Maroon">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> implies <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i/>;<br/>


</div>
