<?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_3.html#V3">continuous</a> <a href="waybel11.html#V4" target="_self">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><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>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Olive">q</font></span>)</span> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">q</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">p</font> </span>}</span> ;<br/>
<a NAME="E1:103"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Olive">q</font></span>)</span> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">q</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">p</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_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="E1:103_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Olive">q</font></span>)</span> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">q</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">p</font> </span>}</span> 
 ;<br/>

<a NAME="E2:103_1"/><b>then </b>
 ex <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">p</font> )
 
;<br/>
<b>hence </b><a NAME="E3:103_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Olive">q</font></span>)</span> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">q</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">p</font> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E3:103"/>
<font color="Maroon">X</font> is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:103_2"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">L</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_2_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="E1:103_2_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:103_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E4:103_2_1"/>
<font color="Maroon">q</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">p</font>
 ;<br/>
<a NAME="E5:103_2_1"/>
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">q</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="waybel11.html#E3:103_2_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:103_2_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


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

<a NAME="E2:103_2"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:103_2_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:103_2_2"/><b>then </b>
 ex <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">p</font> )
 
;<br/>
<b>hence </b><a NAME="E3:103_2_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:103_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T58">SETFAM_1:58</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/>

<b>assume </b><b>that </b><br/><a NAME="E4:103_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E5:103_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:103_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">u</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E8:103_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, , </i>;<br/>
<b>take </b><font color="Maroon">V</font> =  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">u</font>;<br/>

<b>thus </b><a NAME="E9:103_2"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>

<a NAME="E10:103_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">S</font> is <a href="waybel_0.html#V13">upper</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:103_2"/><i><font color="Green">E89</font></i>: 
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">u</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T11">WAYBEL_3:11</a></i>;<br/>
<a NAME="E12:103_2"/>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">u</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#T5" target="_self">Th5</a>, <a class="ref" href="waybel11.html#T6" target="_self">Th6</a>, </i>;<br/>
<b>hence </b><a NAME="E13:103_2"/>
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="waybel11.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:103"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Olive">q</font></span>)</span> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">q</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">p</font> </span>}</span>  is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">p</font>
 ;<br/>


</div>
