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<div class="add">

<b>set </b><font color="Maroon">C</font> =  <a href="yellow_6.html#K13" target="_self">Convergence</a> <font color="Maroon">T</font>;<br/>
<b>thus </b><a NAME="E1:110_1_1"/>
 <a href="yellow_6.html#K13" target="_self">Convergence</a> <font color="Maroon">T</font> is <a href="yellow_6.html#V4" target="_self">(CONSTANTS)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">N</font> be  <a href="yellow_6.html#V1" target="_self">constant</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D23" target="_self">YELLOW_6:def 23</a><br/>

<b>assume </b><a NAME="E1:110_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E2:110_1_1_1"/>
 <a href="yellow_6.html#K5" target="_self">the_value_of</a> <font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">N</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:110_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N</font>,<span class="p2">(<span class="default"><a href="yellow_6.html#K5" target="_self">the_value_of</a> <font color="Maroon">N</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13" target="_self">Convergence</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a>, </i>;<br/>


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

<b>thus </b><a NAME="E2:110_1_1"/>
 <a href="yellow_6.html#K13" target="_self">Convergence</a> <font color="Maroon">T</font> is <a href="yellow_6.html#V5" target="_self">(SUBNETS)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D24" target="_self">YELLOW_6:def 24</a><br/><b>let </b><font color="Maroon">Y</font> be    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">N</font>;<br/>



<b>assume </b><a NAME="E1:110_1_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 ;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>

<a NAME="E2:110_1_1_2"/><i><font color="Green">E41</font></i>: 
 <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E3:110_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13" target="_self">Convergence</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E4:110_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">N</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:110_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13" target="_self">Convergence</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a>, , </i>;<br/>


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

<b>thus </b><a NAME="E3:110_1_1"/>
 <a href="yellow_6.html#K13" target="_self">Convergence</a> <font color="Maroon">T</font> is <a href="yellow_6.html#V6" target="_self">(DIVERGENCE)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D25" target="_self">YELLOW_6:def 25</a><br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:110_1_1_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 ;<br/>

<b>assume </b><a NAME="E2:110_1_1_3"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13" target="_self">Convergence</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E3:110_1_1_3"/><b>then </b>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a>, </i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:110_1_1_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E6:110_1_1_3"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">Z</font> being   <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">Y</font>  holds not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Olive">Z</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a>, <a class="ref" href="yellow_6.html#D5" target="_self">Def5</a></i>;<br/>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<b>thus </b><a NAME="E7:110_1_1_3"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">Z</font> be    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">Y</font>;<br/>

<a NAME="E8:110_1_1_3"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">Z</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:110_1_1_3"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Z</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13" target="_self">Convergence</a> <font color="Maroon">T</font>
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">X</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D26" target="_self">YELLOW_6:def 26</a><br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E4:110_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13" target="_self">Convergence</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E5:110_1_1"/><b>then </b><i><font color="Green">E117</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font> )
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">J</font> be    <a href="yellow_6.html#M3" target="_self">net_set</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E6:110_1_1"/><i><font color="Green">E118</font></i>: 
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13" target="_self">Convergence</a> <font color="Maroon">T</font>
 ;<br/>

<div><i><font color="Green">E119</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><a NAME="E1:110_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13" target="_self">Convergence</a> <font color="Maroon">T</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:110_1_1_4"/>
( <font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E8:110_1_1"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a>, <a class="txt" href="yellow_6.html#E16"><i><font color="Green">Lemma60</font></i></a></i>;<br/>
<a NAME="E9:110_1_1"/>
 <a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">J</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a>, , </i>;<br/>
<b>hence </b><a NAME="E10:110_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">J</font></span>)</span>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13" target="_self">Convergence</a> <font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/>


</div>
