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<div class="add">

<b>let </b><font color="Maroon">L</font> be  <a href="waybel_3.html#V3">continuous</a> <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D25">YELLOW_6:def 25</a><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>assume </b><a NAME="E1:30"/><i><font color="Green">E25</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">L</font> &amp; not <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="waybel28.html#K3" target="_self">lim_inf-Convergence</a> <font color="Maroon">L</font> )
 ;<br/>

<a NAME="E2:30"/><b>then </b>
not 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>
 
<b>by </b><i/>;<br/>
<a NAME="E3:30"/><b>then </b><i><font color="Green">E28</font></i>: 
(  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font> or  ex <font color="Olive">p</font> being <a href="waybel28.html#V1" target="_self">greater_or_equal_to_id</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">N</font>,<font color="Maroon">N</font> st not <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="waybel_9.html#K2">inf</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel28.html#K2" target="_self">*</a> <font color="Olive">p</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:30"/><i><font color="Green">E29</font></i>: 
 <a href="waybel28.html#K3" target="_self">lim_inf-Convergence</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D21">YELLOW_6:def 21</a></i>;<br/>
<b>consider </b><font color="Maroon">N1</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="E6:30"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">N1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E7:30"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N1</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1"/>
( ( not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font> ) or ( not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font> ) or  ex <font color="Olive">M</font> being   <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font> st <br/>( <font color="Olive">M</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font> &amp; not <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="waybel_9.html#K2">inf</a> <font color="Olive">M</font> ) )
 </span><b>by </b><i>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1"><b>suppose </b></a><a NAME="E1:30_1_1"/><i><font color="Green">E31</font></i>: 
( not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font> )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">N'</font> = <font color="Maroon">N</font> as    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font> <b>by </b><i><a class="ref" href="yellow_6.html#T23">YELLOW_6:23</a></i>;<br/><b>take </b>
<font color="Maroon">N'</font>
;<br/><b>thus </b><a NAME="E3:30_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>
 <b>by </b><i/>;<br/><b>given </b><font color="Maroon">M2</font> being    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N'</font><b> such that </b><a NAME="E4:30_1_1"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">M2</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel28.html#K3" target="_self">lim_inf-Convergence</a> <font color="Maroon">L</font>
 ;<br/><a NAME="E5:30_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">M2</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="waybel28.html#T1" target="_self">Th1</a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:30_1_1"/>
<font color="Maroon">M2</font> is    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">M2</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T23">YELLOW_6:23</a></i>;<br/><a NAME="E7:30_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="waybel28.html#E1:30"><i><font color="Green">E25</font></i></a>, </i>;<br/><a NAME="E8:30_1_1"/>
 <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">M2</font>
 <b>by </b><i><a class="ref" href="waybel21.html#T37">WAYBEL21:37</a></i>;<br/><b>hence </b><a NAME="E9:30_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2"><b>suppose </b></a><a NAME="E1:30_1_2"/>
( not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font> )
 ;<br/><div class="add"><a NAME="E2:30_1_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="waybel11.html#R1">is_S-limit_of</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="waybel11.html#D7">WAYBEL11:def 7</a></i>;<br/><a NAME="E3:30_1_2"/><b>then </b>
not <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="waybel11.html#K2">Scott-Convergence</a> <font color="Maroon">L</font>
 <b>by </b><i>, , <a class="ref" href="waybel11.html#D8">WAYBEL11:def 8</a></i>;<br/><b>then consider </b><font color="Maroon">M</font> being    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E5:30_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E6:30_1_2"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">M1</font> being   <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">M</font>  holds not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">M1</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">Scott-Convergence</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="yellow_6.html#D25">YELLOW_6:def 25</a></i>;<br/><b>take </b>
<font color="Maroon">M</font>
;<br/><b>thus </b><a NAME="E7:30_1_2"/>
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/><a NAME="E8:30_1_2"/>
for <font color="Olive">M1</font> being   <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">M</font>  holds not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">M1</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel28.html#K3" target="_self">lim_inf-Convergence</a> <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">M1</font> be    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">M</font>;<br/>

<b>assume </b><a NAME="E1:30_1_2_1"/><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">M1</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel28.html#K3" target="_self">lim_inf-Convergence</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E2:30_1_2_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">M1</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="waybel28.html#T1" target="_self">Th1</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">M11</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="E4:30_1_2_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">M11</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font>
 <b>and </b><br/><a NAME="E5:30_1_2_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M11</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/>
<a NAME="E6:30_1_2_1"/>
<font color="Maroon">M1</font> is    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">M1</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T23">YELLOW_6:23</a></i>;<br/>
<a NAME="E7:30_1_2_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">M1</font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E1:30"><i><font color="Green">E25</font></i></a>, , </i>;<br/>
<a NAME="E8:30_1_2_1"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">M1</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">Scott-Convergence</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:30_1_2_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="waybel11.html#R1">is_S-limit_of</a> <font color="Maroon">M1</font>
 
<b>by </b><i>, , <a class="ref" href="waybel11.html#D8">WAYBEL11:def 8</a></i>;<br/>
<b>hence </b><a NAME="E10:30_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel28.html#E3:30"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel11.html#D7">WAYBEL11:def 7</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:30_1_2"/>
for <font color="Olive">M1</font> being   <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">M</font>  holds not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">M1</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel28.html#K3" target="_self">lim_inf-Convergence</a> <font color="Maroon">L</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_3"><b>suppose </b></a><a NAME="E1:30_1_3"/>
 ex <font color="Olive">M</font> being   <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font> st <br/>( <font color="Olive">M</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font> &amp; not <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="waybel_9.html#K2">inf</a> <font color="Olive">M</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">M</font> being    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E3:30_1_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E4:30_1_3"/><i><font color="Green">E84</font></i>: 
not <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="waybel_9.html#K2">inf</a> <font color="Maroon">M</font>
 ;<br/><b>take </b>
<font color="Maroon">M</font>
;<br/><b>thus </b><a NAME="E5:30_1_3"/>
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/><a NAME="E6:30_1_3"/>
for <font color="Olive">M1</font> being   <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">M</font>  holds not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">M1</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel28.html#K3" target="_self">lim_inf-Convergence</a> <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">M1</font> be    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">M</font>;<br/>

<b>assume </b><a NAME="E1:30_1_3_1"/><i><font color="Green">E86</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">M1</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel28.html#K3" target="_self">lim_inf-Convergence</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E2:30_1_3_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">M1</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="waybel28.html#T1" target="_self">Th1</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:30_1_3_1"/>
<font color="Maroon">M1</font> is    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">M1</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T23">YELLOW_6:23</a></i>;<br/>
<a NAME="E4:30_1_3_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">M1</font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E1:30"><i><font color="Green">E25</font></i></a>, , </i>;<br/>
<a NAME="E5:30_1_3_1"/><i><font color="Green">E89</font></i>: 
 <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">M1</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="waybel21.html#T37">WAYBEL21:37</a></i>;<br/>
<a NAME="E6:30_1_3_1"/>
 <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">M</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="waybel_9.html#K2">inf</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:30_1_3_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="waybel28.html#E3:30"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:30_1_3"/>
for <font color="Olive">M1</font> being   <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">M</font>  holds not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">M1</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel28.html#K3" target="_self">lim_inf-Convergence</a> <font color="Maroon">L</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
