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<b>let </b><font color="Maroon">T</font> be  <a href="waybel_3.html#V3">continuous</a> <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">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="yellow_0.html#V5">meet-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:65"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:65_1"/>
<font color="Maroon">X</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><a NAME="E2:65_1"/><b>then </b>
<font color="Maroon">S</font> is  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="yellow_0.html#V7">infs-inheriting</a> <a href="waybel_0.html#V4">directed-sups-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="waybel21.html#T2" target="_self">Th2</a>, </i>;<br/><b>hence </b><a NAME="E3:65_1"/>
for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>  st <font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">X</font> holds <br/> <a href="waybel11.html#K1">lim_inf</a> <font color="Olive">N</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:65"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>  st <font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">X</font> holds <br/> <a href="waybel11.html#K1">lim_inf</a> <font color="Olive">N</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:65_2"/>
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/><a NAME="E2:65_2"/><b>then </b>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:65_2"/>
 <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E5:65"/><b>then </b>
( <font color="Maroon">S</font> is <a href="yellow_0.html#V7">infs-inheriting</a> &amp; <font color="Maroon">S</font> is <a href="waybel_0.html#V4">directed-sups-inheriting</a> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:65"/><b>then </b>
 ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Olive">X</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:65"/>
<font color="Maroon">X</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i/>;<br/>


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