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

<b>set </b><font color="Maroon">C</font> =  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font>;<br/>
<b>thus </b><a NAME="E1:96_1_1"/>
(  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font> is <a href="yellow_6.html#V4">(CONSTANTS)</a> &amp;  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font> is <a href="yellow_6.html#V5">(SUBNETS)</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D28">YELLOW_6:def 28</a><br/>

<b>thus </b><a NAME="E2:96_1_1"/>
 <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font> is <a href="yellow_6.html#V6">(DIVERGENCE)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_1"><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">R</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">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:96_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E2:96_1_1_1"/><i><font color="Green">E53</font></i>: 
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="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E3:96_1_1_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds <br/> ( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is <a href="waybel_0.html#V1">directed</a> )
 
;<br/>
<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/>
<b>consider </b><font color="Maroon">N</font> being  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:96_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E7:96_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</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">R</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E11:96_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</a></i>;<br/>
<a NAME="E8:96_1_1_1"/>
not <font color="Maroon">p</font> <a href="waybel11.html#R1" target="_self">is_S-limit_of</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E11:96_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="waybel11.html#T4" target="_self">Th4</a>, , </i>;<br/>
<a NAME="E9:96_1_1_1"/><b>then </b>
not <font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1" target="_self">lim_inf</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E11:96_1_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">u</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">p'</font>
 <b>and </b><br/><a NAME="E12:96_1_1_1"/><i><font color="Green">E88</font></i>: 
not <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1" target="_self">lim_inf</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T24">WAYBEL_3:24</a></i>;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : not <font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">u</font> </span>}</span> ;<br/>
<a NAME="E13:96_1_1_1"/>
<font color="Maroon">X</font> <a href="waybel_0.html#R2">is_often_in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : not <font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">u</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_1_1"><b>proof </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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D12">WAYBEL_0:def 12</a><br/>

<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : <font color="Olive">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">C</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">R</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> ;<br/>
<a NAME="E1:96_1_1_1_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : <font color="Olive">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> </span>}</span> ,<font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">R</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> 
 
;<br/>
<a NAME="E2:96_1_1_1_1"/><b>then </b>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : <font color="Olive">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> </span>}</span> ,<font color="Maroon">R</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1" target="_self">lim_inf</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<a NAME="E3:96_1_1_1_1"/><b>then </b>
not <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : <font color="Olive">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> </span>}</span> ,<font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<a NAME="E4:96_1_1_1_1"/><b>then </b>
not <font color="Maroon">u</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : <font color="Olive">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>
<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:96_1_1_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : <font color="Olive">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> </span>}</span> 
 <b>and </b><br/><a NAME="E7:96_1_1_1_1"/><i><font color="Green">E90</font></i>: 
not <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<b>consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E9:96_1_1_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E10:96_1_1_1_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">j</font>
;<br/>

<b>thus </b><a NAME="E11:96_1_1_1_1"/>
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="waybel11.html#T6" target="_self">Th6</a></i>;<br/>

<b>thus </b><a NAME="E12:96_1_1_1_1"/>
<font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : not <font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">u</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="waybel11.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">X</font> <a href="yellow_6.html#K6">"</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : not <font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">u</font> </span>}</span>  as    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="yellow_6.html#T31">YELLOW_6:31</a></i>;<br/>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<b>reconsider </b><font color="Maroon">UN</font> =  <a href="yellow_6.html#K2">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> as  <a href="classes2.html#V1">universal</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E16:96_1_1_1"/><i><font color="Green">E94</font></i>: 
 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">R</font> st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Olive">N</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">N</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UN</font> )
 
<b>by </b><i><a class="txt" href="waybel11.html#E11:96_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</a></i>;<br/>
<a NAME="E17:96_1_1_1"/>
<font color="Maroon">X</font> <a href="yellow_6.html#K6">"</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : not <font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">u</font> </span>}</span>  is    <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D8">YELLOW_6:def 8</a></i>;<br/>
<a NAME="E18:96_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="yellow_6.html#K6">"</a> <span class="p2">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : not <font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">u</font> </span>}</span> </span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E19:96_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UN</font>
 
<b>by </b><i>, <a class="ref" href="classes1.html#D1">CLASSES1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E20:96_1_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</a></i>;<br/>

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

<b>assume </b><a NAME="E21:96_1_1_1"/><i><font color="Green">E95</font></i>: 
<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="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E22:96_1_1_1"/>
 <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</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">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</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/>
<a NAME="E23:96_1_1_1"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">ZZ</font> being  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E25:96_1_1_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">ZZ</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E26:96_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ZZ</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UN</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> =  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ,<font color="Maroon">R</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>verum;<br/>
<b>set </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>] </span>}</span> ;<br/>
<a NAME="E27:96_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S8">DOMAIN_1:sch 8</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>] </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> ;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E30:96_1_1_1"/>
<font color="Maroon">p</font> <a href="waybel11.html#R1" target="_self">is_S-limit_of</a> <font color="Maroon">ZZ</font>
 
<b>by </b><i>, <a class="ref" href="waybel11.html#T5" target="_self">Th5</a>, <a class="ref" href="waybel11.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E31:96_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1" target="_self">lim_inf</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#T6" target="_self">Th6</a>, </i>;<br/>
<b>then consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E33:96_1_1_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E34:96_1_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#D1">WAYBEL_3:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font><b> such that </b><br/><a NAME="E36:96_1_1_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ,<font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="waybel11.html#D1" target="_self">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> ;<br/>
<b>consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font><b> such that </b><br/><a NAME="E39:96_1_1_1"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j'</font> &amp; <font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j'</font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#D5">YELLOW_6:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Z</font>,<font color="Maroon">Y</font><b> such that </b><br/><a NAME="E41:96_1_1_1"/><i><font color="Green">E102</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">Z</font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">Y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E42:96_1_1_1"/>
for <font color="Olive">m</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>  ex <font color="Olive">n</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> st <br/>for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font>  st <font color="Olive">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">p</font> holds <br/><font color="Olive">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">p</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D12">YELLOW_6:def 12</a></i>;<br/>
<a NAME="E43:96_1_1_1"/>
<font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel11.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E44:96_1_1_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">d</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D3" target="_self">Def3</a>, <a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<a NAME="E45:96_1_1_1"/>
<font color="Maroon">Y</font> <a href="waybel_0.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a>, <a class="ref" href="waybel11.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E46:96_1_1_1"/><b>then </b>
<font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : not <font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">u</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E47:96_1_1_1"/><b>then </b>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> &amp; not <font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">u</font> )
 
;<br/>
<b>hence </b><a NAME="E48:96_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="waybel11.html#D2" target="_self">Def2</a>, <a class="ref" href="waybel11.html#T8" target="_self">Th8</a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E3:96_1_1"/>
 <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font> is <a href="yellow_6.html#V7">(ITERATED_LIMITS)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_2"><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">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D26">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">R</font>;<br/>



<b>assume </b><a NAME="E1:96_1_1_2"/><i><font color="Green">E104</font></i>: 
<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="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font>
 ;<br/>

<b>let </b><font color="Maroon">J</font> be    <a href="yellow_6.html#M3">net_set</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<font color="Maroon">R</font>;<br/>

<b>assume </b><a NAME="E2:96_1_1_2"/><i><font color="Green">E105</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">.</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="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E3:96_1_1_2"/><i><font color="Green">E106</font></i>: 
 <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</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">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</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/>
<a NAME="E4:96_1_1_2"/><b>then </b><i><font color="Green">E230</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:96_1_1_2"/>
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">J</font> <a href="yellow_6.html#K8">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_2_1"><b>proof </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="E1:96_1_1_2_1"/>
<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">.</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="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E2:96_1_1_2_1"/>
<font color="Maroon">J</font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:96_1_1_2"/><b>then </b><i><font color="Green">E231</font></i>: 
 <a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="yellow_6.html#T34">YELLOW_6:34</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/>
<b>set </b><font color="Maroon">q</font> =  <a href="waybel11.html#K1" target="_self">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span>;<br/>
<a NAME="E8:96_1_1_2"/>
 <a href="waybel11.html#K1" target="_self">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">p'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:96_1_1_2_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">p'</font>
 ;<br/>

<a NAME="E2:96_1_1_2_2"/><b>then </b><i><font color="Green">E232</font></i>: 
<font color="Maroon">u</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">p'</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/>
<b>set </b><font color="Maroon">T</font> =  <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>,<span class="p1">(<span class="default"><a href="waybel11.html#K5" target="_self">sigma</a> <font color="Maroon">R</font></span>)</span> #);<br/>
<a NAME="E3:96_1_1_2_2"/><i><font color="Green">E233</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>,<span class="p1">(<span class="default"><a href="waybel11.html#K5" target="_self">sigma</a> <font color="Maroon">R</font></span>)</span> #),the <a href="orders_2.html#U1">InternalRel</a> of <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>,<span class="p1">(<span class="default"><a href="waybel11.html#K5" target="_self">sigma</a> <font color="Maroon">R</font></span>)</span> #) #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> #)
 
;<br/>
<a NAME="E4:96_1_1_2_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D27">YELLOW_6:def 27</a></i>;<br/>
<a NAME="E5:96_1_1_2_2"/><b>then </b><i><font color="Green">E234</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>,<span class="p1">(<span class="default"><a href="waybel11.html#K5" target="_self">sigma</a> <font color="Maroon">R</font></span>)</span> #),the <a href="pre_topc.html#U1">topology</a> of <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>,<span class="p1">(<span class="default"><a href="waybel11.html#K5" target="_self">sigma</a> <font color="Maroon">R</font></span>)</span> #) #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font></span>)</span></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font></span>)</span></span>)</span> #)
 
;<br/>
<b>then reconsider </b><font color="Maroon">T</font> =  <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>,<span class="p1">(<span class="default"><a href="waybel11.html#K5" target="_self">sigma</a> <font color="Maroon">R</font></span>)</span> #) as   <a href="waybel_9.html#NM1">TopLattice</a> <b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">T</font> = <font color="Maroon">T</font> as  <a href="lattice3.html#V3">complete</a> <a href="waybel_9.html#NM1">TopLattice</a> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">T</font> = <font color="Maroon">T</font> as  <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_9.html#NM1">TopLattice</a> <b>by </b><i>, </i>;<br/>
<a NAME="E9:96_1_1_2_2"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K5" target="_self">sigma</a> <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="waybel11.html#T2" target="_self">Th2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">T</font> = <font color="Maroon">T</font> as  <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel11.html#V4" target="_self">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">XX</font> = <font color="Maroon">X</font> as   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">JJ</font> = <font color="Maroon">J</font> as    <a href="yellow_6.html#M3">net_set</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XX</font>,<font color="Maroon">T</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">uu</font> = <font color="Maroon">u</font>, <font color="Maroon">qq</font> =  <a href="waybel11.html#K1" target="_self">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span>, <font color="Maroon">pp</font> = <font color="Maroon">p'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/>
<b>set </b><font color="Maroon">N</font> =  <a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">JJ</font>;<br/>
<b>set </b><font color="Maroon">CC</font> =  <a href="yellow_6.html#K13">Convergence</a> <font color="Maroon">T</font>;<br/>
<a NAME="E14:96_1_1_2_2"/>
 <a href="yellow_6.html#K13">Convergence</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="yellow_6.html#K13">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:96_1_1_2_2"/><b>then </b><i><font color="Green">E241</font></i>: 
 <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_6.html#K13">Convergence</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T49">YELLOW_6:49</a></i>;<br/>
<a NAME="E16:96_1_1_2_2"/><i><font color="Green">E242</font></i>: 
<font color="Maroon">uu</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">pp</font>
 
<b>by </b><i>, , <a class="ref" href="waybel11.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E17:96_1_1_2_2"/>
 <a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">JJ</font> <a href="hidden.html#R1">=</a>  <a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E18:96_1_1_2_2"/><b>then </b><i><font color="Green">E243</font></i>: 
<font color="Maroon">qq</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1" target="_self">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">JJ</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E19:96_1_1_2_2"/>
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">XX</font> holds  <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">JJ</font> <a href="yellow_6.html#K8">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">XX</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">Convergence</a> <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_2_2_1"><b>proof </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">XX</font>;<br/>

<b>reconsider </b><font color="Maroon">ii</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E2:96_1_1_2_2_1"/>
<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">.</a> <font color="Maroon">ii</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">ii</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E3:96_1_1_2_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">JJ</font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">XX</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">Convergence</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="waybel11.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:96_1_1_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">JJ</font></span>)</span>,<font color="Maroon">pp</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13">Convergence</a> <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="waybel11.html#T5" target="_self">Th5</a>, <a class="ref" href="yellow_6.html#D26">YELLOW_6:def 26</a></i>;<br/>
<a NAME="E21:96_1_1_2_2"/><b>then </b><i><font color="Green">E244</font></i>: 
<font color="Maroon">pp</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">JJ</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D22">YELLOW_6:def 22</a></i>;<br/>
<a NAME="E22:96_1_1_2_2"/><i><font color="Green">E245</font></i>: 
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">uu</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<a NAME="E23:96_1_1_2_2"/>
<font color="Maroon">pp</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">uu</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E24:96_1_1_2_2"/><b>then </b>
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">uu</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">pp</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D3" target="_self">Def3</a>, <a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/>
<a NAME="E25:96_1_1_2_2"/><b>then </b><i><font color="Green">E246</font></i>: 
 <a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">JJ</font> <a href="waybel_0.html#R1">is_eventually_in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">uu</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D2" target="_self">Def2</a>, <a class="ref" href="yellow_6.html#D18">YELLOW_6:def 18</a></i>;<br/>
<a NAME="E26:96_1_1_2_2"/>
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">uu</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">uu</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T11">WAYBEL_3:11</a></i>;<br/>
<a NAME="E27:96_1_1_2_2"/><b>then </b>
 <a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">JJ</font> <a href="waybel_0.html#R1">is_eventually_in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">uu</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D4" target="_self">Def4</a>, <a class="ref" href="waybel_0.html#T8">WAYBEL_0:8</a></i>;<br/>
<a NAME="E28:96_1_1_2_2"/><b>then </b>
<font color="Maroon">uu</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">qq</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>hence </b><a NAME="E29:96_1_1_2_2"/>
<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1" target="_self">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:96_1_1_2"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">p'</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1" target="_self">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<a NAME="E10:96_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1" target="_self">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a></i>;<br/>
<a NAME="E11:96_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="waybel11.html#R1" target="_self">is_S-limit_of</a>  <a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:96_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K9">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="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font>
 <b>by </b><i>, </i>;<br/>


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


</div>
