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

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

<b>set </b><font color="Maroon">SC</font> =  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">L</font>;<br/>
<b>set </b><font color="Maroon">T</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">L</font></span>)</span>;<br/>
<a NAME="E1:66"/><i><font color="Green">E41</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">L</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">L</font> : for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</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">L</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">L</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/>
<b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:66_1"/>
<font color="Maroon">S</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">L</font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:66_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E4:66_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> holds <br/>for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</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">L</font> holds <br/><font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:66_1"/>
<font color="Maroon">S</font> is <a href="waybel11.html#V1" target="_self">inaccessible_by_directed_joins</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D1" target="_self">WAYBEL11:def 1</a><br/>

<b>assume </b><a NAME="E1:66_1_1"/><i><font color="Green">E63</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:66_1_1"/><i><font color="Green">E64</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> =  <a href="partfun1.html#K6">id</a> <font color="Maroon">D</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">D</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">N</font> =  <a href="waybel11.html#K3" target="_self">Net-Str</a> <font color="Maroon">D</font>,<font color="Maroon">f</font> as  <a href="orders_2.html#V2">reflexive</a> <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#V8">monotone</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font> <b>by </b><i><a class="txt" href="waybel11.html#E1:66"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<a NAME="E5:66_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
 <a href="waybel11.html#K1" target="_self">lim_inf</a> <font color="Maroon">N</font> = 
 <a href="waybel_2.html#K1">sup</a> <font color="Maroon">N</font>
<b>by </b><i/>
<br/>.= 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i><a class="ref" href="waybel_2.html#D1">WAYBEL_2:def 1</a></i>
<br/>.= 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font></span>)</span>,<font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>
<br/>.= 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font>
<b>by </b><i/>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>
;<br/>
<a NAME="E7:66_1_1"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="waybel11.html#R1" target="_self">is_S-limit_of</a> <font color="Maroon">N</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:66_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:66_1_1"/><b>then </b>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">i0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E11:66_1_1"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st <font color="Maroon">i0</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">j</font> holds <br/><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">j0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E13:66_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">j0</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i0</font>
 <b>and </b><br/><a NAME="E14:66_1_1"/>
<font color="Maroon">j0</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i0</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D5">YELLOW_6:def 5</a></i>;<br/>
<a NAME="E15:66_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E16:66_1_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 
<b>by </b><i/>;<br/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j0</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j0</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">j0</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>
<b>hence </b><a NAME="E18:66_1_1"/>
<font color="Maroon">D</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E6:66_1"/>
<font color="Maroon">S</font> is <a href="waybel_0.html#V13">upper</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:66_1_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E2:66_1_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E3:66_1_2"/><i><font color="Green">E96</font></i>: 
 <a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1"><i><font color="Green">Lemma34</font></i></a></i>;<br/>
<a NAME="E4:66_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1" target="_self">lim_inf</a> <span class="p1">(<span class="default"><a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:66_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="waybel11.html#R1" target="_self">is_S-limit_of</a>  <a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:66"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<a NAME="E6:66_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:66_1_2"/><b>then </b>
 <a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">y</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E8:66_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E3:66"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">S</font> is <a href="waybel11.html#V1" target="_self">inaccessible_by_directed_joins</a>
 <b>and </b><br/><a NAME="E4:66"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">S</font> is <a href="waybel_0.html#V13">upper</a>
 ;<br/>

<a NAME="E5:66"/>
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/>for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</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">L</font> holds <br/><font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:66_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E3:66_2"/><i><font color="Green">E100</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E4:66_2"/>
 <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D21">YELLOW_6:def 21</a></i>;<br/>
<a NAME="E5:66_2"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:66"><i><font color="Green">E41</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:66_2"/><b>then </b>
 ex <font color="Olive">N'</font> being <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">N'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">N'</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</a></i>;<br/>
<a NAME="E7:66_2"/><b>then </b>
<font color="Maroon">p</font> <a href="waybel11.html#R1" target="_self">is_S-limit_of</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:66"><i><font color="Green">E41</font></i></a>, , </i>;<br/>
<a NAME="E8:66_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">p'</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1" target="_self">lim_inf</a> <font color="Maroon">N</font>
 
<b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>verum;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> =  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ,<font color="Maroon">L</font>;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>] </span>}</span> ;<br/>
<a NAME="E9:66_2"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S8">DOMAIN_1:sch 8</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>] </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<a NAME="E12:66_2"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, , , <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>
<a NAME="E13:66_2"/><b>then </b>
<font color="Maroon">D</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:66_2"/><b>then </b>
<font color="Maroon">D</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E16:66_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">d</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E18:66_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E20:66_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ,<font color="Maroon">L</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<b>take </b>
<font color="Maroon">j</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a><br/>

<b>let </b><font color="Maroon">i0</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>;<br/>

<a NAME="E22:66_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>assume </b><a NAME="E23:66_2"/>
<font color="Maroon">j</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">i0</font>
 ;<br/>

<a NAME="E24:66_2"/><b>then </b>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> 
 
;<br/>
<a NAME="E25:66_2"/><b>then </b>
<font color="Maroon">d</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i0</font>
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<b>hence </b><a NAME="E26:66_2"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:66"/>
<font color="Maroon">S</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">L</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


</div>
