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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:70"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">N</font> being <a href="waybel_0.html#V10">eventually-directed</a> <a href="waybel_0.html#NM2">net</a> of  <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">X</font> holds <br/> (  <a href="waybel_9.html#R1">ex_sup_of</a> <font color="Olive">N</font> &amp;  <a href="waybel_2.html#K1">sup</a> <font color="Olive">N</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Olive">N</font> )
 ;<br/>

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel25.html#D4" target="_self">WAYBEL25:def 4</a><br/>

<a NAME="E2:70"/><i><font color="Green">E43</font></i>: 
 <a href="waybel_9.html#R1">ex_sup_of</a>  <a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:70"/>
 <a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">X</font></span>)</span> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">D</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="waybel17.html#D4">WAYBEL17:def 4</a></i>;<br/>
<a NAME="E4:70"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="yellow14.html#T2">YELLOW14:2</a></i>;<br/>
<b>hence </b><a NAME="E5:70"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D</font>, <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="waybel_9.html#D3">WAYBEL_9:def 3</a></i>;<br/>

<b>let </b><font color="Maroon">V</font> be  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E6:70"/><i><font color="Green">E47</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/>

<a NAME="E7:70"/><i><font color="Green">E54</font></i>: 
 <a href="waybel_2.html#K1">sup</a> <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E9:70"/>
<font color="Maroon">Z</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="waybel_2.html#K1">sup</a> <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<i><font color="Green">E55</font></i>:  <a href="waybel_2.html#K1">sup</a> <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> = 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i><a class="ref" href="waybel_2.html#D1">WAYBEL_2:def 1</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p2">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span></span>)</span>

;<br/>
<a NAME="E2:70_1"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">X</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">X</font></span>)</span> #)
 
<b>by </b><i/>;<br/>
<a NAME="E3:70_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="tops_3.html#T77">TOPS_3:77</a></i>;<br/>
<b>hence </b><a NAME="E4:70_1"/>
 <a href="waybel_2.html#K1">sup</a> <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">Z</font>
 <b>by </b><i>, <a class="txt" href="waybel25.html#E1:70"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel25.html#E4:70"><i><font color="Green">E44</font></i></a>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a><br/>


</div><b>end;</b></div>
<a NAME="E10:70"/><b>then </b>
 <a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E2:70"><i><font color="Green">E43</font></i></a>, <a class="ref" href="yellow_6.html#D18">YELLOW_6:def 18</a></i>;<br/>
<b>then consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span><b> such that </b><br/><a NAME="E12:70"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span>  st <font color="Maroon">i</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font>;<br/><a NAME="E1:70_2"/>
 <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E2:70_2"/><b>then </b>
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:70_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:70_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E4:70"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:70"/>
<font color="Maroon">D</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


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