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

<span class="kw">set </span><font color="Maroon" title="c2">C</font> =  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font>;<br/>
<span class="kw">thus </span><a NAME="E1:95_1_1"/>
(  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font> is  <a href="yellow_6.html#V4" title="YELLOW_6:attr.4">(CONSTANTS)</a>  &amp;  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font> is  <a href="yellow_6.html#V5" title="YELLOW_6:attr.5">(SUBNETS)</a>  )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="yellow_6.html#D25" title="YELLOW_6:def.25">YELLOW_6:def 25</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font> is  <a href="yellow_6.html#V6" title="YELLOW_6:attr.6">(DIVERGENCE)</a>  &amp;  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font> is  <a href="yellow_6.html#V7" title="YELLOW_6:attr.7">(ITERATED_LIMITS)</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:95_1_1"/>
 <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font> is  <a href="yellow_6.html#V6" title="YELLOW_6:attr.6">(DIVERGENCE)</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font> is  <a href="yellow_6.html#V7" title="YELLOW_6:attr.7">(ITERATED_LIMITS)</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">X</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">R</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="yellow_6.html#D22" title="YELLOW_6:def.22">YELLOW_6:def 22</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  holds <br/>(  not <font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Olive">b<sub>1</sub></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">R</font> or  ex <font color="Olive">b<sub>2</sub></font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c3">X</font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">R</font> &amp; (  for <font color="Olive">b<sub>3</sub></font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive">b<sub>2</sub></font>  holds  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></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">R</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">p</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">R</font> or  ex <font color="Olive">b<sub>1</sub></font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c3">X</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">R</font> &amp; (  for <font color="Olive">b<sub>2</sub></font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive">b<sub>1</sub></font>  holds  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c4">p</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">R</font> ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:95_1_1_1"/><span class="lab"><font color="Green" title="E64">A1</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">and </span><br/><a NAME="E2:95_1_1_1"/><span class="lab"><font color="Green" title="E65">A2</font></span>: 
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">p</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">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c3">X</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">R</font> &amp; (  for <font color="Olive">b<sub>2</sub></font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive">b<sub>1</sub></font>  holds  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c4">p</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">R</font> ) )</span><br/>

<a NAME="E3:95_1_1_1"/><span class="lab"><font color="Green" title="E66">A3</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> holds <br/> (  not  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  )
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">p9</font> = <font color="Maroon" title="c4">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">N</font> being   <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a>  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E6:95_1_1_1"/><span class="lab"><font color="Green" title="E67">A4</font></span>: 
<font color="Maroon" title="c6">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">and </span><br/><a NAME="E7:95_1_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_1"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="ref" href="yellow_6.html#D11" title="YELLOW_6:def.11">YELLOW_6:def 11</a></span>;<br/>
<a NAME="E8:95_1_1_1"/>
 not <font color="Maroon" title="c4">p</font> <a href="waybel11.html#R1" title="WAYBEL11:pred.1">is_S-limit_of</a> <font color="Maroon" title="c6">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_1"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E2:95_1_1_1"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="txt" href="#E6:95_1_1_1"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="ref" href="waybel11.html#D8" target="_self" title="WAYBEL11:def.8">Def8</a></span>;<br/>
<a NAME="E9:95_1_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c4">p</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="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:95_1_1_1"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="ref" href="waybel11.html#D7" target="_self" title="WAYBEL11:def.7">Def7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E11:95_1_1_1"/><span class="lab"><font color="Green" title="E68">A5</font></span>: 
<font color="Maroon" title="c7">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">p9</font>
 <span class="kw">and </span><br/><a NAME="E12:95_1_1_1"/><span class="lab"><font color="Green" title="E69">A6</font></span>: 
 not <font color="Maroon" title="c7">u</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="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:95_1_1_1"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="ref" href="waybel_3.html#T24" title="WAYBEL_3:th.24">WAYBEL_3:24</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">A</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> :  not <font color="Olive" title="b1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">u</font> </span> } </span> ;<br/>
<a NAME="E13:95_1_1_1"/>
<font color="Maroon" title="c3">X</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> :  not <font color="Olive" title="b1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">u</font> </span> } </span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">X</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_0.html#D12" title="WAYBEL_0:def.12">WAYBEL_0:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X</font> st <br/>( <font color="Maroon" title="c9">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">a</font> where <font color="Olive" title="b2">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> :  not <font color="Olive" title="b2">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">u</font> </span> } </span>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">B</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">X</font> : <font color="Olive" title="b1">j</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">i</font> </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">C</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<span class="p3"> { <span class="default"> <span class="p4">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">k</font></span>)</span> where <font color="Olive" title="b2">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">X</font> : <font color="Olive" title="b2">k</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">j</font> </span> } </span> ,<font color="Maroon" title="c1">R</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">X</font> : verum </span> } </span> ;<br/>
<a NAME="E1:95_1_1_1_1"/>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">X</font> : <font color="Olive" title="b1">j</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">i</font> </span> } </span> ,<font color="Maroon" title="c1">R</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<span class="p3"> { <span class="default"> <span class="p4">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">k</font></span>)</span> where <font color="Olive" title="b2">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">X</font> : <font color="Olive" title="b2">k</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">j</font> </span> } </span> ,<font color="Maroon" title="c1">R</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">X</font> : verum </span> } </span> 
 
;<br/>
<a NAME="E2:95_1_1_1_1"/><span class="kw">then </span>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">X</font> : <font color="Olive" title="b1">j</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">i</font> </span> } </span> ,<font color="Maroon" title="c1">R</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="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T22" title="YELLOW_2:th.22">YELLOW_2:22</a></span>;<br/>
<a NAME="E3:95_1_1_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c7">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">X</font> : <font color="Olive" title="b1">j</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">i</font> </span> } </span> ,<font color="Maroon" title="c1">R</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:95_1_1_1"><span class="lab"><font color="Green" title="E69">A6</font></span></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></span>;<br/>
<a NAME="E4:95_1_1_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c7">u</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">X</font> : <font color="Olive" title="b1">j</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">i</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E6:95_1_1_1_1"/><span class="lab"><font color="Green" title="E70">A7</font></span>: 
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">X</font> : <font color="Olive" title="b1">j</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">i</font> </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E7:95_1_1_1_1"/><span class="lab"><font color="Green" title="E71">A8</font></span>: 
 not <font color="Maroon" title="c7">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">X</font><span class="kw"> such that </span><br/><a NAME="E9:95_1_1_1_1"/><span class="lab"><font color="Green" title="E72">A9</font></span>: 
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">j</font>
 <span class="kw">and </span><br/><a NAME="E10:95_1_1_1_1"/><span class="lab"><font color="Green" title="E73">A10</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:95_1_1_1_1"><span class="lab"><font color="Green" title="E70">A7</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">j</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> :  not <font color="Olive" title="b1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">u</font> </span> } </span>  )</span><br/>

<span class="kw">thus </span><a NAME="E11:95_1_1_1_1"/>
<font color="Maroon" title="c9">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:95_1_1_1_1"><span class="lab"><font color="Green" title="E73">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> :  not <font color="Olive" title="b1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">u</font> </span> } </span> </span><br/>

<span class="kw">thus </span><a NAME="E12:95_1_1_1_1"/>
<font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> :  not <font color="Olive" title="b1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">u</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:95_1_1_1_1"><span class="lab"><font color="Green" title="E71">A8</font></span></a>, <a class="txt" href="#E9:95_1_1_1_1"><span class="lab"><font color="Green" title="E72">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">Y</font> = <font color="Maroon" title="c3">X</font> <a href="yellow_6.html#K5" title="YELLOW_6:func.5">"</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> :  not <font color="Olive" title="b1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">u</font> </span> } </span>  as    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c3">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#T22" title="YELLOW_6:th.22">YELLOW_6:22</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">Y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">R</font> &amp; (  for <font color="Olive">b<sub>1</sub></font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c9">Y</font>  holds  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c4">p</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">R</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">UN</font> =  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> as   <a href="classes2.html#V1" title="CLASSES2:attr.1">universal</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E16:95_1_1_1"/><span class="lab"><font color="Green" title="E70">A11</font></span>: 
 ex <font color="Olive" title="b1">N</font> being   <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a>  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">N</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">UN</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_1"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="ref" href="yellow_6.html#D11" title="YELLOW_6:def.11">YELLOW_6:def 11</a></span>;<br/>
<a NAME="E17:95_1_1_1"/>
<font color="Maroon" title="c3">X</font> <a href="yellow_6.html#K5" title="YELLOW_6:func.5">"</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> :  not <font color="Olive" title="b1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">u</font> </span> } </span>  is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D6" title="YELLOW_6:def.6">YELLOW_6:def 6</a></span>;<br/>
<a NAME="E18:95_1_1_1"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_6.html#K5" title="YELLOW_6:func.5">"</a> <span class="p2"> { <span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> :  not <font color="Olive" title="b1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">u</font> </span> } </span> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></span>;<br/>
<a NAME="E19:95_1_1_1"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">UN</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:95_1_1_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:95_1_1_1"/>
<font color="Maroon" title="c9">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D11" title="YELLOW_6:def.11">YELLOW_6:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c9">Y</font>  holds  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c4">p</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">R</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">Z</font> be    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c9">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">Z</font>,<font color="Maroon" title="c4">p</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">R</font></span><br/>

<span class="kw">assume </span><a NAME="E21:95_1_1_1"/><span class="lab"><font color="Green" title="E71">A12</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">Z</font>,<font color="Maroon" title="c4">p</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">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E22:95_1_1_1"/>
 <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</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#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">R</font></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D18" title="YELLOW_6:def.18">YELLOW_6:def 18</a></span>;<br/>
<a NAME="E23:95_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A13</font></span>: 
<font color="Maroon" title="c11">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:95_1_1_1"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">ZZ</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">R</font><span class="kw"> such that </span><br/><a NAME="E25:95_1_1_1"/><span class="lab"><font color="Green" title="E73">A14</font></span>: 
<font color="Maroon" title="c12">ZZ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Z</font>
 <span class="kw">and </span><br/><a NAME="E26:95_1_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">ZZ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">UN</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D11" title="YELLOW_6:def.11">YELLOW_6:def 11</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Z</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> =  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c11">Z</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Z</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c1">R</font> </span> } </span> ,<font color="Maroon" title="c1">R</font>);<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>verum;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">D</font> = <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>) where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Z</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] </span> } </span> ;<br/>
<a NAME="E27:95_1_1_1"/>
<span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>) where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Z</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] </span> } </span>  is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S8" title="DOMAIN_1:sch.8">DOMAIN_1:sch 8</a>();<br/></span>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">D</font> = <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>) where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Z</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] </span> } </span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">D</font> = <font color="Maroon" title="c14">D</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#T30" target="_self" title="WAYBEL11:th.30">Th30</a></span>;<br/>
<a NAME="E30:95_1_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="waybel11.html#R1" title="WAYBEL11:pred.1">is_S-limit_of</a> <font color="Maroon" title="c12">ZZ</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:95_1_1_1"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="txt" href="#E23:95_1_1_1"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="txt" href="#E25:95_1_1_1"><span class="lab"><font color="Green" title="E73">A14</font></span></a>, <a class="ref" href="waybel11.html#D8" target="_self" title="WAYBEL11:def.8">Def8</a></span>;<br/>
<a NAME="E31:95_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">p</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="c11">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:95_1_1_1"><span class="lab"><font color="Green" title="E73">A14</font></span></a>, <a class="ref" href="waybel11.html#D7" target="_self" title="WAYBEL11:def.7">Def7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E33:95_1_1_1"/><span class="lab"><font color="Green" title="E74">A15</font></span>: 
<font color="Maroon" title="c16">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">D</font>
 <span class="kw">and </span><br/><a NAME="E34:95_1_1_1"/><span class="lab"><font color="Green" title="E75">A16</font></span>: 
<font color="Maroon" title="c7">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:95_1_1_1"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="waybel_3.html#D1" title="WAYBEL_3:def.1">WAYBEL_3:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Z</font><span class="kw"> such that </span><br/><a NAME="E36:95_1_1_1"/><span class="lab"><font color="Green" title="E76">A17</font></span>: 
<font color="Maroon" title="c16">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c11">Z</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Z</font> : <font color="Olive" title="b1">k</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c17">j</font> </span> } </span> ,<font color="Maroon" title="c1">R</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:95_1_1_1"><span class="lab"><font color="Green" title="E74">A15</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">j9</font> = <font color="Maroon" title="c17">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Z</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Z</font><span class="kw"> such that </span><br/><a NAME="E39:95_1_1_1"/><span class="lab"><font color="Green" title="E77">A18</font></span>: 
<font color="Maroon" title="c19">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c18">j9</font>
 <span class="kw">and </span><br/><a NAME="E40:95_1_1_1"/>
<font color="Maroon" title="c19">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c18">j9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D3" title="YELLOW_6:def.3">YELLOW_6:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c11">Z</font>,<font color="Maroon" title="c9">Y</font><span class="kw"> such that </span><br/><a NAME="E42:95_1_1_1"/><span class="lab"><font color="Green" title="E78">A19</font></span>: 
 the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c11">Z</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c9">Y</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">f</font>
 <span class="kw">and </span><br/><a NAME="E43:95_1_1_1"/>
 for <font color="Olive" title="b1">m</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">Y</font>  ex <font color="Olive" title="b2">n</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Z</font> st <br/> for <font color="Olive" title="b3">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Z</font>  st <font color="Olive" title="b2">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b1">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c20">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b3">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D9" title="YELLOW_6:def.9">YELLOW_6:def 9</a></span>;<br/>
<a NAME="E44:95_1_1_1"/>
<font color="Maroon" title="c11">Z</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c11">Z</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Z</font> : <font color="Olive" title="b1">k</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c17">j</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:95_1_1_1"><span class="lab"><font color="Green" title="E77">A18</font></span></a></span>;<br/>
<a NAME="E45:95_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A20</font></span>: 
<font color="Maroon" title="c16">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">Z</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c19">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:95_1_1_1"><span class="lab"><font color="Green" title="E76">A17</font></span></a>, <a class="ref" href="yellow_2.html#T22" title="YELLOW_2:th.22">YELLOW_2:22</a></span>;<br/>
<a NAME="E46:95_1_1_1"/>
<font color="Maroon" title="c9">Y</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c19">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Z</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c19">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:95_1_1_1"><span class="lab"><font color="Green" title="E78">A19</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>;<br/>
<a NAME="E47:95_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">Z</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> :  not <font color="Olive" title="b1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">u</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#T35" target="_self" title="WAYBEL11:th.35">Th35</a></span>;<br/>
<a NAME="E48:95_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Z</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c19">i</font> &amp;  not <font color="Olive" title="b1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">u</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E49:95_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:95_1_1_1"><span class="lab"><font color="Green" title="E75">A16</font></span></a>, <a class="txt" href="#E45:95_1_1_1"><span class="lab"><font color="Green" title="E79">A20</font></span></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:95_1_1"/>
 <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font> is  <a href="yellow_6.html#V7" title="YELLOW_6:attr.7">(ITERATED_LIMITS)</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">X</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">R</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="yellow_6.html#D23" title="YELLOW_6:def.23">YELLOW_6:def 23</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  holds <br/>(  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Olive">b<sub>1</sub></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">R</font> or  for <font color="Olive">b<sub>2</sub></font> being    <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">R</font>  holds <br/>(  ex <font color="Olive">b<sub>3</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X</font> st  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="yellow_6.html#K7" title="YELLOW_6:func.7">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></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">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></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">R</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">p</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">R</font> or  for <font color="Olive">b<sub>1</sub></font> being    <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">R</font>  holds <br/>(  ex <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X</font> st  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="yellow_6.html#K7" title="YELLOW_6:func.7">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></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">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Maroon" title="c4">p</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">R</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:95_1_1_2"/><span class="lab"><font color="Green" title="E64">A21</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">p</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">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">R</font>  holds <br/>(  ex <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X</font> st  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="yellow_6.html#K7" title="YELLOW_6:func.7">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></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">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Maroon" title="c4">p</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">R</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">J</font> be    <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X</font> st  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">J</font> <a href="yellow_6.html#K7" title="YELLOW_6:func.7">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></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">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c5">J</font></span>)</span>,<font color="Maroon" title="c4">p</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">R</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:95_1_1_2"/><span class="lab"><font color="Green" title="E65">A22</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">X</font> holds  <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">J</font> <a href="yellow_6.html#K7" title="YELLOW_6:func.7">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span></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">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c5">J</font></span>)</span>,<font color="Maroon" title="c4">p</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">R</font></span><br/>

<a NAME="E3:95_1_1_2"/><span class="lab"><font color="Green" title="E66">A23</font></span>: 
 <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</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#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">R</font></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D18" title="YELLOW_6:def.18">YELLOW_6:def 18</a></span>;<br/>
<a NAME="E4:95_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A24</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_2"><span class="lab"><font color="Green" title="E64">A21</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E5:95_1_1_2"/>
 for <font color="Olive" title="b1">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">X</font> holds  <font color="Maroon" title="c5">J</font> <a href="yellow_6.html#K7" title="YELLOW_6:func.7">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">J</font> <a href="yellow_6.html#K7" title="YELLOW_6:func.7">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">R</font></span><br/>

<a NAME="E1:95_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">J</font> <a href="yellow_6.html#K7" title="YELLOW_6:func.7">.</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span></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">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:95_1_1_2"><span class="lab"><font color="Green" title="E65">A22</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:95_1_1_2_1"/>
<font color="Maroon" title="c5">J</font> <a href="yellow_6.html#K7" title="YELLOW_6:func.7">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:95_1_1_2"><span class="lab"><font color="Green" title="E66">A23</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:95_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A25</font></span>: 
 <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c5">J</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:95_1_1_2"><span class="lab"><font color="Green" title="E67">A24</font></span></a>, <a class="ref" href="yellow_6.html#T25" title="YELLOW_6:th.25">YELLOW_6:25</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">p9</font> = <font color="Maroon" title="c4">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">q</font> =  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c5">J</font></span>)</span>;<br/>
<a NAME="E8:95_1_1_2"/>
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c5">J</font></span>)</span> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c6">p9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c6">p9</font> or <font color="Maroon" title="c8">u</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c5">J</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:95_1_1_2_2"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c6">p9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">u</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c5">J</font></span>)</span></span><br/>

<a NAME="E2:95_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A26</font></span>: 
<font color="Maroon" title="c8">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c6">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#T7" title="WAYBEL_3:th.7">WAYBEL_3:7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">T</font> =  <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>, the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>,<span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">R</font></span>)</span> #);<br/>
<a NAME="E3:95_1_1_2_2"/><span class="lab"><font color="Green" title="E70">A27</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>, the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>,<span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">R</font></span>)</span> #), the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>, the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>,<span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">R</font></span>)</span> #) #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>, the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> #)
 
;<br/>
<a NAME="E4:95_1_1_2_2"/><span class="lab"><font color="Green" title="E71">A28</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K13" title="YELLOW_6:func.13">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D24" title="YELLOW_6:def.24">YELLOW_6:def 24</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">T</font> =  <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>, the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>,<span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">R</font></span>)</span> #) as   <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:95_1_1_2_2"><span class="lab"><font color="Green" title="E70">A27</font></span></a>, <a class="txt" href="waybel11.html#E52"><span class="lab"><font color="Green" title="E43">Lm3</font></span></a>, <a class="txt" href="waybel11.html#E58"><span class="lab"><font color="Green" title="E49">Lm9</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">T</font> = <font color="Maroon" title="c10">T</font> as   <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:95_1_1_2_2"><span class="lab"><font color="Green" title="E70">A27</font></span></a>, <a class="txt" href="waybel11.html#E71"><span class="lab"><font color="Green" title="E62">Lm22</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">T</font> = <font color="Maroon" title="c11">T</font> as   <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>   <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:95_1_1_2_2"><span class="lab"><font color="Green" title="E70">A27</font></span></a>, <a class="txt" href="waybel11.html#E72"><span class="lab"><font color="Green" title="E63">Lm23</font></span></a></span>;<br/>
<a NAME="E8:95_1_1_2_2"/>
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c12">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c12">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:95_1_1_2_2"><span class="lab"><font color="Green" title="E70">A27</font></span></a>, <a class="txt" href="waybel11.html#E70"><span class="lab"><font color="Green" title="E61">Lm21</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">T</font> = <font color="Maroon" title="c12">T</font> as   <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>   <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>   <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>  <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#T37" target="_self" title="WAYBEL11:th.37">Th37</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">XX</font> = <font color="Maroon" title="c3">X</font> as   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c13">T</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">JJ</font> = <font color="Maroon" title="c5">J</font> as    <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">XX</font>,<font color="Maroon" title="c13">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:95_1_1_2_2"><span class="lab"><font color="Green" title="E70">A27</font></span></a>, <a class="txt" href="waybel11.html#E67"><span class="lab"><font color="Green" title="E58">Lm18</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">uu</font> = <font color="Maroon" title="c8">u</font>, <font color="Maroon" title="c17">qq</font> =  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c5">J</font></span>)</span>, <font color="Maroon" title="c18">pp</font> = <font color="Maroon" title="c6">p9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">T</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">N</font> =  <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c15">JJ</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">CC</font> =  <a href="yellow_6.html#K12" title="YELLOW_6:func.12">Convergence</a> <font color="Maroon" title="c13">T</font>;<br/>
<a NAME="E13:95_1_1_2_2"/>
 <a href="yellow_6.html#K12" title="YELLOW_6:func.12">Convergence</a> <font color="Maroon" title="c13">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K12" title="YELLOW_6:func.12">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K13" title="YELLOW_6:func.13">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:95_1_1_2_2"><span class="lab"><font color="Green" title="E71">A28</font></span></a>, <a class="txt" href="waybel11.html#E57"><span class="lab"><font color="Green" title="E48">Lm8</font></span></a></span>;<br/>
<a NAME="E14:95_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A29</font></span>: 
 <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K12" title="YELLOW_6:func.12">Convergence</a> <font color="Maroon" title="c13">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#T40" title="YELLOW_6:th.40">YELLOW_6:40</a></span>;<br/>
<a NAME="E15:95_1_1_2_2"/><span class="lab"><font color="Green" title="E73">A30</font></span>: 
<font color="Maroon" title="c16">uu</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c18">pp</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:95_1_1_2_2"><span class="lab"><font color="Green" title="E69">A26</font></span></a>, <a class="txt" href="#E3:95_1_1_2_2"><span class="lab"><font color="Green" title="E70">A27</font></span></a>, <a class="txt" href="waybel11.html#E64"><span class="lab"><font color="Green" title="E55">Lm15</font></span></a></span>;<br/>
<a NAME="E16:95_1_1_2_2"/><span class="lab"><font color="Green" title="E74">A31</font></span>: 
<font color="Maroon" title="c17">qq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c15">JJ</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:95_1_1_2_2"><span class="lab"><font color="Green" title="E70">A27</font></span></a>, <a class="txt" href="waybel11.html#E65"><span class="lab"><font color="Green" title="E56">Lm16</font></span></a>, <a class="txt" href="waybel11.html#E66"><span class="lab"><font color="Green" title="E57">Lm17</font></span></a></span>;<br/>
<a NAME="E17:95_1_1_2_2"/>
 for <font color="Olive" title="b1">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">XX</font> holds  <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">JJ</font> <a href="yellow_6.html#K7" title="YELLOW_6:func.7">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">XX</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span></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="yellow_6.html#K12" title="YELLOW_6:func.12">Convergence</a> <font color="Maroon" title="c13">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">XX</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">JJ</font> <a href="yellow_6.html#K7" title="YELLOW_6:func.7">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">XX</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c21">i</font></span>)</span></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="yellow_6.html#K12" title="YELLOW_6:func.12">Convergence</a> <font color="Maroon" title="c13">T</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c22">ii</font> = <font color="Maroon" title="c21">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">X</font> ;<br/>
<a NAME="E2:95_1_1_2_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">J</font> <a href="yellow_6.html#K7" title="YELLOW_6:func.7">.</a> <font color="Maroon" title="c22">ii</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c22">ii</font></span>)</span></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">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:95_1_1_2"><span class="lab"><font color="Green" title="E65">A22</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:95_1_1_2_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">JJ</font> <a href="yellow_6.html#K7" title="YELLOW_6:func.7">.</a> <font color="Maroon" title="c21">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">XX</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c21">i</font></span>)</span></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="yellow_6.html#K12" title="YELLOW_6:func.12">Convergence</a> <font color="Maroon" title="c13">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:95_1_1_2_2"><span class="lab"><font color="Green" title="E72">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E18:95_1_1_2_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c15">JJ</font></span>)</span>,<font color="Maroon" title="c18">pp</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="yellow_6.html#K12" title="YELLOW_6:func.12">Convergence</a> <font color="Maroon" title="c13">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:95_1_1_2"><span class="lab"><font color="Green" title="E64">A21</font></span></a>, <a class="txt" href="#E14:95_1_1_2_2"><span class="lab"><font color="Green" title="E72">A29</font></span></a>, <a class="ref" href="yellow_6.html#D23" title="YELLOW_6:def.23">YELLOW_6:def 23</a></span>;<br/>
<a NAME="E19:95_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A32</font></span>: 
<font color="Maroon" title="c18">pp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K10" title="YELLOW_6:func.10">Lim</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c15">JJ</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D19" title="YELLOW_6:def.19">YELLOW_6:def 19</a></span>;<br/>
<a NAME="E20:95_1_1_2_2"/>
<font color="Maroon" title="c18">pp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c16">uu</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:95_1_1_2_2"><span class="lab"><font color="Green" title="E73">A30</font></span></a></span>;<br/>
<a NAME="E21:95_1_1_2_2"/><span class="kw">then </span>
 <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c16">uu</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c18">pp</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#T36" target="_self" title="WAYBEL11:th.36">Th36</a>, <a class="ref" href="connsp_2.html#T3" title="CONNSP_2:th.3">CONNSP_2:3</a></span>;<br/>
<a NAME="E22:95_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A33</font></span>: 
 <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c15">JJ</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c16">uu</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:95_1_1_2_2"><span class="lab"><font color="Green" title="E75">A32</font></span></a>, <a class="ref" href="yellow_6.html#D15" title="YELLOW_6:def.15">YELLOW_6:def 15</a></span>;<br/>
<a NAME="E23:95_1_1_2_2"/>
 <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c16">uu</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c16">uu</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#T11" title="WAYBEL_3:th.11">WAYBEL_3:11</a></span>;<br/>
<a NAME="E24:95_1_1_2_2"/><span class="kw">then </span>
 <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c15">JJ</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c16">uu</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:95_1_1_2_2"><span class="lab"><font color="Green" title="E76">A33</font></span></a>, <a class="ref" href="waybel_0.html#T8" title="WAYBEL_0:th.8">WAYBEL_0:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:95_1_1_2_2"/>
<font color="Maroon" title="c8">u</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> <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c5">J</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:95_1_1_2_2"><span class="lab"><font color="Green" title="E70">A27</font></span></a>, <a class="txt" href="#E16:95_1_1_2_2"><span class="lab"><font color="Green" title="E74">A31</font></span></a>, <a class="txt" href="waybel11.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="waybel11.html#T18" target="_self" title="WAYBEL11:th.18">Th18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:95_1_1_2"/><span class="kw">then </span>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c6">p9</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c5">J</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></span>;<br/>
<a NAME="E10:95_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">p</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> <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c5">J</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#D5" title="WAYBEL_3:def.5">WAYBEL_3:def 5</a></span>;<br/>
<a NAME="E11:95_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">p</font> <a href="waybel11.html#R1" title="WAYBEL11:pred.1">is_S-limit_of</a>  <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c5">J</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#D7" target="_self" title="WAYBEL11:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:95_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c5">J</font></span>)</span>,<font color="Maroon" title="c4">p</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">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:95_1_1_2"><span class="lab"><font color="Green" title="E68">A25</font></span></a>, <a class="ref" href="waybel11.html#D8" target="_self" title="WAYBEL11:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
