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

<b>let </b><font color="Maroon">R1</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">R2</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:93"/><i><font color="Green">E41</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R1</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R1</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R2</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R2</font> #)
 ;<br/>

<b>set </b><font color="Maroon">T1</font> =  <a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p1">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R1</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">T2</font> =  <a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p1">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R2</font></span>)</span>;<br/>
<a NAME="E2:93"/><i><font color="Green">E53</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R1</font> : for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R1</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> holds <br/>for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R1</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">N</font>,<font color="Olive">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R1</font> holds <br/><font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Olive">V</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_6.html#D27">YELLOW_6:def 27</a></i>;<br/>
<a NAME="E3:93"/><i><font color="Green">E54</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R2</font> : for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R2</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> holds <br/>for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R2</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">N</font>,<font color="Olive">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R2</font> holds <br/><font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Olive">V</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_6.html#D27">YELLOW_6:def 27</a></i>;<br/>
<a NAME="E4:93"/>
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R2</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:93_1"/>
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R2</font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:93_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R1</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">V1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R1</font><b> such that </b><br/><a NAME="E3:93_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V1</font>
 <b>and </b><br/><a NAME="E4:93_1_1"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R1</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> holds <br/>for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R1</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">N</font>,<font color="Olive">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R1</font> holds <br/><font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">V1</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E3:93_1_1"><i><font color="Green">E63</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">V2</font> = <font color="Maroon">V1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R2</font> <b>by </b><i/>;<br/>
<a NAME="E6:93_1_1"/>
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R2</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font> holds <br/>for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R2</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">N</font>,<font color="Olive">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R2</font> holds <br/><font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">V2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R2</font>;<br/>

<b>assume </b><a NAME="E1:93_1_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R1</font> <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">N2</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R2</font>;<br/>

<b>reconsider </b><font color="Maroon">N1</font> = <font color="Maroon">N2</font> as   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R1</font> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E4:93_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R2</font>
 ;<br/>

<a NAME="E5:93_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">p1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:93_1_1_1"/><b>then </b>
<font color="Maroon">N1</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">V1</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:93_1_1_1"/>
<font color="Maroon">N2</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">V2</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:93_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel11.html#E4:93_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="waybel11.html#T4" target="_self">Th4</a></i>;<br/>


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

<b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:93_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R2</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">V1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R2</font><b> such that </b><br/><a NAME="E4:93_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V1</font>
 <b>and </b><br/><a NAME="E5:93_1"/><i><font color="Green">E90</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R2</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> holds <br/>for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R2</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">N</font>,<font color="Olive">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R2</font> holds <br/><font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">V1</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">V2</font> = <font color="Maroon">V1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R1</font> <b>by </b><i/>;<br/>
<a NAME="E7:93_1"/>
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R1</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font> holds <br/>for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R1</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">N</font>,<font color="Olive">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R1</font> holds <br/><font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">V2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R1</font>;<br/>

<b>assume </b><a NAME="E1:93_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R2</font> <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">N2</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R1</font>;<br/>

<b>reconsider </b><font color="Maroon">N1</font> = <font color="Maroon">N2</font> as   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R2</font> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E4:93_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R1</font>
 ;<br/>

<a NAME="E5:93_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">p1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:93_1_2"/><b>then </b>
<font color="Maroon">N1</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">V1</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:93_1_2"/>
<font color="Maroon">N2</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">V2</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:93_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R1</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel11.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:93"/>
 <a href="waybel11.html#K5" target="_self">sigma</a> <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K5" target="_self">sigma</a> <font color="Maroon">R2</font>
 ;<br/>


</div>
