<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">S</font> be  <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>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:95"/><i><font color="Green" title="E48">A1</font></i>: 
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> )
 <b>and </b><br/><a NAME="E2:95"/><i><font color="Green" title="E49">A2</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>
 ;<br/>

<a NAME="E3:95"/>
for <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> holds   <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c3">n</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">X</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">N</font> =  <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">n</font> #);<br/>
<a NAME="E2:95_1"/>
 <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">n</font> #) is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>
 
<b>by </b><i><a class="ref" href="waybel_2.html#T20" title="WAYBEL_2:th.20">WAYBEL_2:20</a></i>;<br/>
<a NAME="E3:95_1"/><b>then </b><i><font color="Green" title="E50">A3</font></i>: 
 <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a>  <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">n</font> #)
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:95"><i><font color="Green" title="E48">A1</font></i></a></i>;<br/>
<a NAME="E4:95_1"/>
 <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 <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">n</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/>
<b>hence </b><a NAME="E5:95_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E3:95_1"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="waybel_9.html#D3" target="_self" title="WAYBEL_9:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:95"/>
<font color="Maroon" title="c1">S</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#T75" title="WAYBEL_0:th.75">WAYBEL_0:75</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_2.html#D7" title="WAYBEL_2:def.7">WAYBEL_2:def 7</a><br/>

<a NAME="E5:95"/>
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><br/> for <font color="Olive" title="b2">M</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="b2">M</font> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a> holds <br/><font color="Olive" title="b1">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Olive" title="b2">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <font color="Olive" title="b2">M</font>,<font color="Maroon" title="c1">S</font></span>)</span></span>)</span></span>)</span>
 
<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="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">S</font>;<br/>



<b>assume </b><a NAME="E1:95_2"/><i><font color="Green" title="E50">A4</font></i>: 
<font color="Maroon" title="c3">M</font> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">xM</font> = <font color="Maroon" title="c2">x</font> <a href="waybel_9.html#K1" title="WAYBEL_9:func.1">"/\"</a> <font color="Maroon" title="c3">M</font>;<br/>
<a NAME="E2:95_2"/><i><font color="Green" title="E51">A5</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="waybel_9.html#K1" title="WAYBEL_9:func.1">"/\"</a> <font color="Maroon" title="c3">M</font> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:95_2"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="waybel_2.html#T27" title="WAYBEL_2:th.27">WAYBEL_2:27</a></i>;<br/>
<a NAME="E3:95_2"/><i><font color="Green" title="E52">A6</font></i>: 
<font color="Maroon" title="c2">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>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E2:95"><i><font color="Green" title="E49">A2</font></i></a></i>;<br/>
<a NAME="E4:95_2"/><i><font color="Green" title="E53">A7</font></i>: 
 <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Maroon" title="c3">M</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="c3">M</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:95"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="waybel_9.html#E1:95_2"><i><font color="Green" title="E50">A4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">M1</font> = <font color="Maroon" title="c3">M</font> as  <a href="yellow_6.html#V3" title="YELLOW_6:attr.3">convergent</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="ref" href="yellow_6.html#D19" title="YELLOW_6:def.19">YELLOW_6:def 19</a></i>;<br/>
<a NAME="E6:95_2"/><i><font color="Green" title="E54">A8</font></i>: 
 <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_9.html#K1" title="WAYBEL_9:func.1">"/\"</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_9.html#K1" title="WAYBEL_9:func.1">"/\"</a> <font color="Maroon" title="c3">M</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:95"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="waybel_9.html#E2:95_2"><i><font color="Green" title="E51">A5</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">xM</font> = <font color="Maroon" title="c2">x</font> <a href="waybel_9.html#K1" title="WAYBEL_9:func.1">"/\"</a> <font color="Maroon" title="c3">M</font> as  <a href="yellow_6.html#V3" title="YELLOW_6:attr.3">convergent</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="ref" href="yellow_6.html#D19" title="YELLOW_6:def.19">YELLOW_6:def 19</a></i>;<br/>
<a NAME="E8:95_2"/><i><font color="Green" title="E55">A9</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K12" title="YELLOW_6:func.12">lim</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="waybel_9.html#K1" title="WAYBEL_9:func.1">"/\"</a> <font color="Maroon" title="c5">M1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E3:95_2"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="waybel_9.html#T26" target="_self" title="WAYBEL_9:th.26">Th26</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Maroon" title="c3">M</font></span>)</span> = 
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K12" title="YELLOW_6:func.12">lim</a> <font color="Maroon" title="c5">M1</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_9.html#E4:95_2"><i><font color="Green" title="E53">A7</font></i></a>, <a class="ref" href="yellow_6.html#D20" title="YELLOW_6:def.20">YELLOW_6:def 20</a></i>
<br/>.= 
 <a href="yellow_6.html#K12" title="YELLOW_6:func.12">lim</a> <font color="Maroon" title="c6">xM</font>
<b>by </b><i><a class="txt" href="waybel_9.html#E8:95_2"><i><font color="Green" title="E55">A9</font></i></a>, <a class="ref" href="yellow_6.html#D20" title="YELLOW_6:def.20">YELLOW_6:def 20</a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
<b>by </b><i><a class="txt" href="waybel_9.html#E6:95_2"><i><font color="Green" title="E54">A8</font></i></a>, <a class="ref" href="yellow_6.html#D20" title="YELLOW_6:def.20">YELLOW_6:def 20</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><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="c6">xM</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c1">S</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_2.html#T23" title="WAYBEL_2:th.23">WAYBEL_2:23</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:95"/>
<font color="Maroon" title="c1">S</font> is <a href="waybel_2.html#V1" title="WAYBEL_2:attr.1">satisfying_MC</a>
 <b>by </b><i><a class="ref" href="waybel_9.html#T9" target="_self" title="WAYBEL_9:th.9">Th9</a></i>;<br/>


</div>
