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<b>let </b><font color="Maroon" title="c1">S</font> be  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> <a href="compts_1.html#NV6" title="COMPTS_1:NV.6">Hausdorff</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">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Olive" title="b1">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> ) implies 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">S</font>  st <font color="Olive" title="b1">N</font> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a> holds <br/>(  <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a> <font color="Olive" title="b1">N</font> &amp;  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Olive" title="b1">N</font> ) )</span><br/>

<b>assume </b><a NAME="E1:95"/><i><font color="Green" title="E49">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Olive" title="b1">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</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">S</font>  st <font color="Olive" title="b1">N</font> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a> holds <br/>(  <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a> <font color="Olive" title="b1">N</font> &amp;  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Olive" title="b1">N</font> )</span><br/>

<b>let </b><font color="Maroon" title="c2">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">N</font> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a> implies (  <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">N</font> &amp;  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Maroon" title="c2">N</font> ) )</span><br/>

<b>assume </b><a NAME="E2:95"/><i><font color="Green" title="E50">A2</font></i>: 
<font color="Maroon" title="c2">N</font> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">N</font> &amp;  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Maroon" title="c2">N</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c3">c</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E4:95"/><i><font color="Green" title="E51">A3</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Maroon" title="c2">N</font>
 <b>by </b><i><a class="ref" href="waybel_9.html#T30" target="_self" title="WAYBEL_9:th.30">Th30</a></i>;<br/>
<a NAME="E5:95"/><i><font color="Green" title="E52">A4</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Maroon" title="c2">N</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:95"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="waybel_9.html#E2:95"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="waybel_9.html#E4:95"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="waybel_9.html#T35" target="_self" title="WAYBEL_9:th.35">Th35</a></i>;<br/>
<b>thus </b><a NAME="E6:95"/>
 <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">N</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Maroon" title="c2">N</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c4">X</font> =  <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="Maroon" title="c2">N</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">d</font> = <font color="Maroon" title="c3">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E2:95_1"/><i><font color="Green" title="E53">A5</font></i>: 
 <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="Maroon" title="c2">N</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c5">d</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:95"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="waybel_9.html#E2:95"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="waybel_9.html#E4:95"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="waybel_9.html#E48"><i><font color="Green" title="E41">Lm5</font></i></a></i>;<br/>
<a NAME="E3:95_1"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</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="Maroon" title="c2">N</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c5">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">b</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:95"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="waybel_9.html#E4:95"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="waybel_9.html#E49"><i><font color="Green" title="E42">Lm6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:95_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <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="Maroon" title="c2">N</font>,<font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E2:95_1"><i><font color="Green" title="E53">A5</font></i></a>, <a class="ref" href="yellow_0.html#T15" title="YELLOW_0:th.15">YELLOW_0:15</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_9.html#D3" target="_self" title="WAYBEL_9:def.3">WAYBEL_9:def 3</a> <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>

<a NAME="E7:95"/>
for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">c</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Maroon" title="c2">N</font> &amp; <font color="Olive" title="b2">d</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Maroon" title="c2">N</font> holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">c</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Maroon" title="c2">N</font> &amp; <font color="Maroon" title="c5">d</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Maroon" title="c2">N</font> implies <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:95_2"/><i><font color="Green" title="E53">A6</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Maroon" title="c2">N</font>
 <b>and </b><br/><a NAME="E2:95_2"/><i><font color="Green" title="E54">A7</font></i>: 
<font color="Maroon" title="c5">d</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Maroon" title="c2">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font></span><br/>

<b>thus </b><font color="Maroon" title="c4">c</font> = 
 <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Maroon" title="c2">N</font>
<b>by </b><i><a class="txt" href="waybel_9.html#E1:95"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="waybel_9.html#E2:95"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="waybel_9.html#E1:95_2"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="waybel_9.html#T35" target="_self" title="WAYBEL_9:th.35">Th35</a></i>
<br/>.= 
<font color="Maroon" title="c5">d</font>
<b>by </b><i><a class="txt" href="waybel_9.html#E1:95"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="waybel_9.html#E2:95"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="waybel_9.html#E2:95_2"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="waybel_9.html#T35" target="_self" title="WAYBEL_9:th.35">Th35</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>hence </b><a NAME="E8:95"/>
 <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Maroon" title="c2">N</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E4:95"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="waybel_9.html#E5:95"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="waybel_9.html#T33" target="_self" title="WAYBEL_9:th.33">Th33</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>
