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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">N</font> being <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">N</font> &amp; ( for <font color="Olive">M</font> being   <a href="yellow_6.html#M2">subnet</a> of <font color="Olive">N</font> holds  <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Olive">M</font> ) );<br/>
<b>consider </b><font color="Maroon">C</font> being    <a href="relset_1.html#M2">Relation</a> of  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:24_1_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">N'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font><br/> for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">N'</font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">N'</font>,<font color="Olive">x</font>] )
 <b>from </b><i><a class="ref" href="relset_1.html#S2">RELSET_1:sch 2</a>();<br/></i>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as    <a href="yellow_6.html#M4">Convergence-Class</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="yellow_6.html#D21">YELLOW_6:def 21</a></i>;<br/>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E4:24_1_1"/>
<font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">N1</font> = <font color="Maroon">N</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font> ;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>thus </b><a NAME="E6:24_1_1"/>
( <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> <font color="Maroon">C</font> implies for <font color="Olive">M</font> being   <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font> holds  <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Olive">M</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_1_1"/><i><font color="Green">E28</font></i>: 
<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> <font color="Maroon">C</font>
 ;<br/>

<b>let </b><font color="Maroon">M</font> be    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font>;<br/>

<b>consider </b><font color="Maroon">N2</font> being  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:24_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">N1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N2</font>
 <b>and </b><br/><a NAME="E4:24_1_1_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">M</font> being   <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N2</font> holds  <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Olive">M</font>
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E5:24_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">M</font>
 <b>by </b><i>, </i>;<br/>


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

<b>assume </b><a NAME="E7:24_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">M</font> being   <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font> holds  <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Olive">M</font>
 ;<br/>

<b>consider </b><font color="Maroon">N2</font> being  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:24_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">N2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N1</font>
 <b>and </b><br/><a NAME="E10:24_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N2</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">L</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</a></i>;<br/>
<b>thus </b><a NAME="E11:24_1_1"/>
<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> <font color="Maroon">C</font>
 <b>by </b><i>, , </i>;<br/>


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