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

<b>let </b><font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c3">C2</font> be    <a href="yellow_6.html#M4" title="YELLOW_6:mode.4">Convergence-Class</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:24_1_2"/><i><font color="Green" title="E17">A7</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">L</font>  st <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font> holds <br/>for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">N</font>,<font color="Olive" title="b2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C1</font> iff for <font color="Olive" title="b3">M</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b1">N</font> holds  <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Olive" title="b3">M</font> )
 <b>and </b><br/><a NAME="E2:24_1_2"/><i><font color="Green" title="E18">A8</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">L</font>  st <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font> holds <br/>for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">N</font>,<font color="Olive" title="b2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C2</font> iff for <font color="Olive" title="b3">M</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b1">N</font> holds  <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Olive" title="b3">M</font> )
 ;<br/>

<a NAME="E3:24_1_2"/><i><font color="Green" title="E19">A9</font></i>: 
<font color="Maroon" title="c2">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D21" title="YELLOW_6:def.21">YELLOW_6:def 21</a></i>;<br/>
<a NAME="E4:24_1_2"/><i><font color="Green" title="E20">A10</font></i>: 
<font color="Maroon" title="c3">C2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D21" title="YELLOW_6:def.21">YELLOW_6:def 21</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">Ns</font>, <font color="Maroon" title="c5">xs</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D2" title="RELAT_1:def.2">RELAT_1:def 2</a><br/>

<b>thus </b><a NAME="E5:24_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">Ns</font>,<font color="Maroon" title="c5">xs</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C1</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">Ns</font>,<font color="Maroon" title="c5">xs</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_2_1"/><i><font color="Green" title="E21">A11</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">Ns</font>,<font color="Maroon" title="c5">xs</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C1</font>
 ;<br/>

<a NAME="E2:24_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">Ns</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E3:24_1_2"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">N</font> being  <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E4:24_1_2_1"/><i><font color="Green" title="E22">A12</font></i>: 
<font color="Maroon" title="c6">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Ns</font>
 <b>and </b><br/><a NAME="E5:24_1_2_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D14" title="YELLOW_6:def.14">YELLOW_6:def 14</a></i>;<br/>
<a NAME="E6:24_1_2_1"/><i><font color="Green" title="E23">A13</font></i>: 
<font color="Maroon" title="c6">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E3:24_1_2"><i><font color="Green" title="E19">A9</font></i></a>, <a class="txt" href="waybel28.html#E1:24_1_2_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="waybel28.html#E4:24_1_2_1"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c5">xs</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel28.html#E3:24_1_2"><i><font color="Green" title="E19">A9</font></i></a>, <a class="txt" href="waybel28.html#E1:24_1_2_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E8:24_1_2_1"/>
for <font color="Olive" title="b1">M</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c6">N</font> holds  <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Olive" title="b1">M</font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E1:24_1_2"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="waybel28.html#E1:24_1_2_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="waybel28.html#E4:24_1_2_1"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="waybel28.html#E6:24_1_2_1"><i><font color="Green" title="E23">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:24_1_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">Ns</font>,<font color="Maroon" title="c5">xs</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C2</font>
 <b>by </b><i><a class="txt" href="waybel28.html#E2:24_1_2"><i><font color="Green" title="E18">A8</font></i></a>, <a class="txt" href="waybel28.html#E4:24_1_2_1"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="waybel28.html#E6:24_1_2_1"><i><font color="Green" title="E23">A13</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E6:24_1_2"/><i><font color="Green" title="E21">A14</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">Ns</font>,<font color="Maroon" title="c5">xs</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C2</font>
 ;<br/>

<a NAME="E7:24_1_2"/><b>then </b>
<font color="Maroon" title="c4">Ns</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E4:24_1_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">N</font> being  <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E9:24_1_2"/><i><font color="Green" title="E22">A15</font></i>: 
<font color="Maroon" title="c6">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Ns</font>
 <b>and </b><br/><a NAME="E10:24_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D14" title="YELLOW_6:def.14">YELLOW_6:def 14</a></i>;<br/>
<a NAME="E11:24_1_2"/><i><font color="Green" title="E23">A16</font></i>: 
<font color="Maroon" title="c6">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E4:24_1_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="waybel28.html#E6:24_1_2"><i><font color="Green" title="E21">A14</font></i></a>, <a class="txt" href="waybel28.html#E9:24_1_2"><i><font color="Green" title="E22">A15</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c5">xs</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel28.html#E4:24_1_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="waybel28.html#E6:24_1_2"><i><font color="Green" title="E21">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E13:24_1_2"/>
for <font color="Olive" title="b1">M</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c6">N</font> holds  <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Olive" title="b1">M</font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E2:24_1_2"><i><font color="Green" title="E18">A8</font></i></a>, <a class="txt" href="waybel28.html#E6:24_1_2"><i><font color="Green" title="E21">A14</font></i></a>, <a class="txt" href="waybel28.html#E9:24_1_2"><i><font color="Green" title="E22">A15</font></i></a>, <a class="txt" href="waybel28.html#E11:24_1_2"><i><font color="Green" title="E23">A16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:24_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">Ns</font>,<font color="Maroon" title="c5">xs</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C1</font>
 <b>by </b><i><a class="txt" href="waybel28.html#E1:24_1_2"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="waybel28.html#E9:24_1_2"><i><font color="Green" title="E22">A15</font></i></a>, <a class="txt" href="waybel28.html#E11:24_1_2"><i><font color="Green" title="E23">A16</font></i></a></i>;<br/>


</div>
