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<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/><b>let </b><font color="Maroon" title="c2">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D25" title="YELLOW_6:def.25">YELLOW_6:def 25</a><br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c2">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> &amp; not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">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>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> )
 ;<br/>

<a NAME="E2:28"/><b>then </b>
not 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="c2">N</font> holds  <font color="Maroon" title="c3">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="ref" href="waybel28.html#D3" target="_self" title="WAYBEL28:def.3">Def3</a></i>;<br/>
<a NAME="E3:28"/><b>then </b><i><font color="Green" title="E18">A2</font></i>: 
(  not <font color="Maroon" title="c3">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="Maroon" title="c2">N</font> or  ex <font color="Olive" title="b1">p</font> being <a href="waybel28.html#V1" title="WAYBEL28:attr.1">greater_or_equal_to_id</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c2">N</font> st not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel28.html#K2" title="WAYBEL28:func.2">*</a> <font color="Olive" title="b1">p</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel28.html#T15" target="_self" title="WAYBEL28:th.15">Th15</a></i>;<br/>
<a NAME="E4:28"/><i><font color="Green" title="E19">A3</font></i>: 
 <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</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>consider </b><font color="Maroon" title="c4">N1</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="E6:28"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c4">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font>
 <b>and </b><br/><a NAME="E7:28"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N1</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="txt" href="waybel28.html#E1:28"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="yellow_6.html#D14" title="YELLOW_6:def.14">YELLOW_6:def 14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1"/>
( ( not <font color="Maroon" title="c3">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="Maroon" title="c2">N</font> &amp; <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c2">N</font> ) or ( not <font color="Maroon" title="c3">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="Maroon" title="c2">N</font> &amp; not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c2">N</font> ) or  ex <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="c2">N</font> st <br/>( <font color="Olive" title="b1">M</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> &amp; not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <font color="Olive" title="b1">M</font> ) )
 </span><b>by </b><i><a class="txt" href="waybel28.html#E1:28"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="waybel28.html#E3:28"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="waybel28.html#T11" target="_self" title="WAYBEL28:th.11">Th11</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><b>suppose </b></a><a NAME="E1:28_1_1"/><i><font color="Green" title="E21">A5</font></i>: 
( not <font color="Maroon" title="c3">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="Maroon" title="c2">N</font> &amp; <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c2">N</font> )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c5">N'</font> = <font color="Maroon" title="c2">N</font> as    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font> <b>by </b><i><a class="ref" href="yellow_6.html#T23" title="YELLOW_6:th.23">YELLOW_6:23</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">N'</font>
;<br/><b>thus </b><a NAME="E3:28_1_1"/>
<font color="Maroon" title="c5">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#E1:28"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/><b>given </b><font color="Maroon" title="c6">M2</font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">N'</font><b> such that </b><a NAME="E5:28_1_1"/><i><font color="Green" title="E22">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c3">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>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font>
 ;<br/><a NAME="E6:28_1_1"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c6">M2</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#E4:28"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="waybel28.html#E5:28_1_1"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E7:28_1_1"/>
<font color="Maroon" title="c6">M2</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c6">M2</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T23" title="YELLOW_6:th.23">YELLOW_6:23</a></i>;<br/><a NAME="E8:28_1_1"/><b>then </b><i><font color="Green" title="E24">A8</font></i>: 
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c6">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="waybel28.html#E5:28_1_1"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="waybel28.html#E6:28_1_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="waybel28.html#D3" target="_self" title="WAYBEL28:def.3">Def3</a></i>;<br/><a NAME="E9:28_1_1"/>
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c2">N</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c6">M2</font>
 <b>by </b><i><a class="ref" href="waybel21.html#T37" title="WAYBEL21:th.37">WAYBEL21:37</a></i>;<br/><b>hence </b><a NAME="E10:28_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel28.html#E1:28_1_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="waybel28.html#E8:28_1_1"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="yellow_0.html#D3" title="YELLOW_0:def.3">YELLOW_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2"><b>suppose </b></a><a NAME="E1:28_1_2"/>
( not <font color="Maroon" title="c3">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="Maroon" title="c2">N</font> &amp; not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c2">N</font> )
 ;<br/><div class="add"><a NAME="E2:28_1_2"/><b>then </b>
not <font color="Maroon" title="c3">x</font> <a href="waybel11.html#R1" title="WAYBEL11:pred.1">is_S-limit_of</a> <font color="Maroon" title="c2">N</font>
 <b>by </b><i><a class="ref" href="waybel11.html#D7" title="WAYBEL11:def.7">WAYBEL11:def 7</a></i>;<br/><a NAME="E3:28_1_2"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">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>  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel28.html#E1:28"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="waybel28.html#E6:28"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="waybel11.html#D8" title="WAYBEL11:def.8">WAYBEL11:def 8</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">M</font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E5:28_1_2"/><i><font color="Green" title="E21">A9</font></i>: 
<font color="Maroon" title="c5">M</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>and </b><br/><a NAME="E6:28_1_2"/><i><font color="Green" title="E22">A10</font></i>: 
for <font color="Olive" title="b1">M1</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">M</font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">M1</font>,<font color="Maroon" title="c3">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>  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel28.html#E1:28"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="yellow_6.html#D25" title="YELLOW_6:def.25">YELLOW_6:def 25</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">M</font>
;<br/><b>thus </b><a NAME="E7:28_1_2"/>
<font color="Maroon" title="c5">M</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#E5:28_1_2"><i><font color="Green" title="E21">A9</font></i></a></i>;<br/><a NAME="E8:28_1_2"/>
for <font color="Olive" title="b1">M1</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">M</font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">M1</font>,<font color="Maroon" title="c3">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>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">M1</font> be    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">M</font>;<br/>

<b>assume </b><a NAME="E1:28_1_2_1"/><i><font color="Green" title="E23">A11</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">M1</font>,<font color="Maroon" title="c3">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>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font>
 ;<br/>

<a NAME="E2:28_1_2_1"/><b>then </b><i><font color="Green" title="E24">A12</font></i>: 
<font color="Maroon" title="c6">M1</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#E4:28"><i><font color="Green" title="E19">A3</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="c7">M11</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="E4:28_1_2_1"/><i><font color="Green" title="E25">A13</font></i>: 
<font color="Maroon" title="c7">M11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M1</font>
 <b>and </b><br/><a NAME="E5:28_1_2_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">M11</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="E6:28_1_2_1"/>
<font color="Maroon" title="c6">M1</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c6">M1</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T23" title="YELLOW_6:th.23">YELLOW_6:23</a></i>;<br/>
<a NAME="E7:28_1_2_1"/><b>then </b><i><font color="Green" title="E26">A14</font></i>: 
<font color="Maroon" title="c3">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="Maroon" title="c6">M1</font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E1:28_1_2_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="waybel28.html#E2:28_1_2_1"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="waybel28.html#D3" target="_self" title="WAYBEL28:def.3">Def3</a></i>;<br/>
<a NAME="E8:28_1_2_1"/>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">M1</font>,<font color="Maroon" title="c3">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>  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E6:28_1_2"><i><font color="Green" title="E22">A10</font></i></a></i>;<br/>
<a NAME="E9:28_1_2_1"/><b>then </b>
not <font color="Maroon" title="c3">x</font> <a href="waybel11.html#R1" title="WAYBEL11:pred.1">is_S-limit_of</a> <font color="Maroon" title="c6">M1</font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E2:28_1_2_1"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="waybel28.html#E4:28_1_2_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="waybel11.html#D8" title="WAYBEL11:def.8">WAYBEL11:def 8</a></i>;<br/>
<b>hence </b><a NAME="E10:28_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel28.html#E7:28_1_2_1"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="waybel11.html#D7" title="WAYBEL11:def.7">WAYBEL11:def 7</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:28_1_2"/>
for <font color="Olive" title="b1">M1</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">M</font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">M1</font>,<font color="Maroon" title="c3">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>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_3"><b>suppose </b></a><a NAME="E1:28_1_3"/>
 ex <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="c2">N</font> st <br/>( <font color="Olive" title="b1">M</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> &amp; not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <font color="Olive" title="b1">M</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">M</font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E3:28_1_3"/><i><font color="Green" title="E21">A15</font></i>: 
<font color="Maroon" title="c5">M</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>and </b><br/><a NAME="E4:28_1_3"/><i><font color="Green" title="E22">A16</font></i>: 
not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <font color="Maroon" title="c5">M</font>
 ;<br/><b>take </b>
<font color="Maroon" title="c5">M</font>
;<br/><b>thus </b><a NAME="E5:28_1_3"/>
<font color="Maroon" title="c5">M</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#E3:28_1_3"><i><font color="Green" title="E21">A15</font></i></a></i>;<br/><a NAME="E6:28_1_3"/>
for <font color="Olive" title="b1">M1</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">M</font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">M1</font>,<font color="Maroon" title="c3">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>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">M1</font> be    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">M</font>;<br/>

<b>assume </b><a NAME="E1:28_1_3_1"/><i><font color="Green" title="E23">A17</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">M1</font>,<font color="Maroon" title="c3">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>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font>
 ;<br/>

<a NAME="E2:28_1_3_1"/><b>then </b><i><font color="Green" title="E24">A18</font></i>: 
<font color="Maroon" title="c6">M1</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#E4:28"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:28_1_3_1"/>
<font color="Maroon" title="c6">M1</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c6">M1</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T23" title="YELLOW_6:th.23">YELLOW_6:23</a></i>;<br/>
<a NAME="E4:28_1_3_1"/><b>then </b><i><font color="Green" title="E25">A19</font></i>: 
<font color="Maroon" title="c3">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="Maroon" title="c6">M1</font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E1:28_1_3_1"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="waybel28.html#E2:28_1_3_1"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="waybel28.html#D3" target="_self" title="WAYBEL28:def.3">Def3</a></i>;<br/>
<a NAME="E5:28_1_3_1"/><i><font color="Green" title="E26">A20</font></i>: 
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c6">M1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c5">M</font>
 
<b>by </b><i><a class="ref" href="waybel21.html#T37" title="WAYBEL21:th.37">WAYBEL21:37</a></i>;<br/>
<a NAME="E6:28_1_3_1"/>
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c5">M</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="waybel_9.html#K2" title="WAYBEL_9:func.2">inf</a> <font color="Maroon" title="c5">M</font>
 
<b>by </b><i><a class="ref" href="waybel28.html#T1" target="_self" title="WAYBEL28:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E7:28_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel28.html#E4:28_1_3"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="waybel28.html#E4:28_1_3_1"><i><font color="Green" title="E25">A19</font></i></a>, <a class="txt" href="waybel28.html#E5:28_1_3_1"><i><font color="Green" title="E26">A20</font></i></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:28_1_3"/>
for <font color="Olive" title="b1">M1</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">M</font>  holds not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">M1</font>,<font color="Maroon" title="c3">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>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
