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<b>let </b><font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel11.html#V4" target="_self">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:97"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E2:97"/><i><font color="Green">E53</font></i>: 
 <a href="yellow_6.html#K13">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T53">YELLOW_6:53</a></i>;<br/>
<b>consider </b><font color="Maroon">M</font> being  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:97"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E5:97"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</a></i>;<br/>
<a NAME="E6:97"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> #) <a href="hidden.html#R1">=</a>  <a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p1">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:97"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="yellow_6.html#K13">Convergence</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="yellow_6.html#K13">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">T</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E8:97"/>
( <font color="Maroon">x</font> <a href="waybel11.html#R1" target="_self">is_S-limit_of</a> <font color="Maroon">N</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">N</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_1"/>
<font color="Maroon">x</font> <a href="waybel11.html#R1" target="_self">is_S-limit_of</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:97_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13">Convergence</a> <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="waybel11.html#T4" target="_self">Th4</a>, , , </i>;<br/>
<b>hence </b><a NAME="E3:97_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D22">YELLOW_6:def 22</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E9:97"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E10:97"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="waybel11.html#T4" target="_self">Th4</a>, , , <a class="ref" href="yellow_6.html#D22">YELLOW_6:def 22</a></i>;<br/>
<b>hence </b><a NAME="E11:97"/>
<font color="Maroon">x</font> <a href="waybel11.html#R1" target="_self">is_S-limit_of</a> <font color="Maroon">N</font>
 <b>by </b><i>, , </i>;<br/>


</div>
