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<b>let </b><font color="Maroon" title="c1">T</font> be  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a> <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">S</font> &amp;  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X</font> holds <br/>( <font color="Olive" title="b2">X</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> iff for <font color="Olive" title="b3">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b3">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Olive" title="b2">X</font> holds <br/> <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Olive" title="b3">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X</font> )</span><br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a> <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> &amp;  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> holds <br/>( <font color="Olive" title="b1">X</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> iff for <font color="Olive" title="b2">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Olive" title="b1">X</font> holds <br/> <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Olive" title="b2">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> )</span><br/><b>let </b><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> &amp;  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies ( <font color="Maroon" title="c3">X</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> iff for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c3">X</font> holds <br/> <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> ) )</span><br/>





<b>assume </b><a NAME="E1:65"/><i><font color="Green" title="E42">A1</font></i>: 
( <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> &amp;  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">X</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> iff for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c3">X</font> holds <br/> <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c3">X</font> holds <br/> <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> ) implies <font color="Maroon" title="c3">X</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:65_1"/>
<font color="Maroon" title="c3">X</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c3">X</font> holds <br/> <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font></span><br/><a NAME="E2:65_1"/><b>then </b>
<font color="Maroon" title="c2">S</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a> <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a> <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="waybel21.html#E1:65"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="waybel21.html#T48" target="_self" title="WAYBEL21:th.48">Th48</a>, <a class="ref" href="waybel21.html#T49" target="_self" title="WAYBEL21:th.49">Th49</a></i>;<br/><b>hence </b><a NAME="E3:65_1"/>
for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c3">X</font> holds <br/> <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="waybel21.html#E1:65"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="waybel21.html#T51" target="_self" title="WAYBEL21:th.51">Th51</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:65"/>
for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c3">X</font> holds <br/> <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">X</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/>

<a NAME="E4:65"/><b>then </b>
for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Olive" title="b1">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> holds <br/> <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="waybel21.html#E1:65"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="waybel21.html#T27" target="_self" title="WAYBEL21:th.27">Th27</a></i>;<br/>
<a NAME="E5:65"/><b>then </b>
( <font color="Maroon" title="c2">S</font> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E1:65"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="waybel21.html#T52" target="_self" title="WAYBEL21:th.52">Th52</a>, <a class="ref" href="waybel21.html#T53" target="_self" title="WAYBEL21:th.53">Th53</a></i>;<br/>
<a NAME="E6:65"/><b>then </b>
 ex <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b1">X</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 
<b>by </b><i><a class="ref" href="waybel21.html#T50" target="_self" title="WAYBEL21:th.50">Th50</a></i>;<br/>
<b>hence </b><a NAME="E7:65"/>
<font color="Maroon" title="c3">X</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="waybel21.html#E1:65"><i><font color="Green" title="E42">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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