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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="compts_1.html#NV4" title="COMPTS_1:NV.4">Hausdorff</a>   <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K10" title="YELLOW_6:func.10">Lim</a> <font color="Olive" title="b1">N</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:95"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K10" title="YELLOW_6:func.10">Lim</a> <font color="Olive" title="b1">N</font> )</span><br/>

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K10" title="YELLOW_6:func.10">Lim</a> <font color="Maroon" title="c2">N</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:95"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K10" title="YELLOW_6:func.10">Lim</a> <font color="Maroon" title="c2">N</font> )</span><br/>

<a NAME="E3:95"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
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_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <font color="Maroon" title="c4">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="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:95_1"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:95_1"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
<font color="Maroon" title="c4">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c3">c</font> = 
 <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Maroon" title="c2">N</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E2:95"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E1:95_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="waybel_9.html#T35" target="_self" title="WAYBEL_9:th.35">Th35</a></span>
<br/>.= 
<font color="Maroon" title="c4">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E2:95"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E2:95_1"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="waybel_9.html#T35" target="_self" title="WAYBEL_9:th.35">Th35</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:95"/><span class="lab"><font color="Green" title="E50">A6</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_9.html#T30" target="_self" title="WAYBEL_9:th.30">Th30</a></span>;<br/>
<span class="kw">thus </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K10" title="YELLOW_6:func.10">Lim</a> <font color="Maroon" title="c2">N</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c4">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/>
<span class="kw">set </span><font color="Maroon" title="c5">X</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">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/>
<a NAME="E2:95_2"/>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">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="c4">d</font> &amp; ( 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#K2" title="RELSET_1:func.2">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="c4">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">b</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E2:95"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E5:95"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="txt" href="waybel_9.html#E48"><span class="lab"><font color="Green" title="E39">Lm5</font></span></a>, <a class="txt" href="waybel_9.html#E49"><span class="lab"><font color="Green" title="E40">Lm6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:95_2"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T15" title="YELLOW_0:th.15">YELLOW_0:15</a></span>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E7:95"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E2:95"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E5:95"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="ref" href="waybel_9.html#T35" target="_self" title="WAYBEL_9:th.35">Th35</a></span>;<br/>
<span class="kw">hence </span><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#K10" title="YELLOW_6:func.10">Lim</a> <font color="Maroon" title="c2">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:95"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="txt" href="#E3:95"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="waybel_9.html#T33" target="_self" title="WAYBEL_9:th.33">Th33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
