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

<b>let </b><font color="Maroon">L1</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon">T</font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">L1</font>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>;<br/>





<b>reconsider </b><font color="Maroon">b1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : verum </span>}</span>  as    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="waybel11.html#T45">WAYBEL11:45</a></i>;<br/>
<b>set </b><font color="Maroon">b2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">u</font></span>)</span></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> </span>}</span> ;<br/>
<a NAME="E2:30"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">u</font></span>)</span></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</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">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:30_1"/>
<font color="Maroon">z</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> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">u</font></span>)</span></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:30_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">u</font></span>)</span>
 <b>and </b><br/><a NAME="E4:30_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font>
 ;<br/>
<b>thus </b><a NAME="E5:30_1"/>
<font color="Maroon">z</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">T</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">b2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">u</font></span>)</span></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E4:30"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">b2</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:30_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:30_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">u</font></span>)</span>
 <b>and </b><br/><a NAME="E4:30_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font>
 ;<br/>
<a NAME="E5:30_2"/>
 <a href="waybel_3.html#K2">wayabove</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">u</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="waybel11.html#T36">WAYBEL11:36</a></i>;<br/>
<b>hence </b><a NAME="E6:30_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:30_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E2:30_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:30_3"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font>
 <b>and </b><br/><a NAME="E5:30_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E6:30_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T31">YELLOW_9:31</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E8:30_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E9:30_3"/>
<font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="yellow_8.html#T19">YELLOW_8:19</a></i>;<br/><b>then consider </b><font color="Maroon">D</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E11:30_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E12:30_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E13:30_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T31">YELLOW_9:31</a></i>;<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/><a NAME="E15:30_3"/>
<font color="Maroon">D</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="yellow_8.html#T19">YELLOW_8:19</a></i>;<br/><b>then consider </b><font color="Maroon">E</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E17:30_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font>
 <b>and </b><br/><a NAME="E18:30_3"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>and </b><br/><a NAME="E19:30_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T31">YELLOW_9:31</a></i>;<br/><b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E21:30_3"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">E</font> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">u</font> =  <a href="waybel_3.html#K2">wayabove</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">D</font></span>)</span>;<br/><b>thus </b><a NAME="E22:30_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b2</font>
 <b>by </b><i/>;<br/><a NAME="E23:30_3"/>
<font color="Maroon">z</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">a1</font>
 <b>by </b><i>, , <a class="ref" href="waybel_3.html#T8">WAYBEL_3:8</a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E25:30_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">z</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E26:30_3"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">y</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#T53">WAYBEL_4:53</a></i>;<br/><a NAME="E27:30_3"/><i><font color="Green">E87</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">D</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/><a NAME="E28:30_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T8">WAYBEL_3:8</a></i>;<br/><a NAME="E29:30_3"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">D</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/><a NAME="E30:30_3"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">D</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">a1</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T2">WAYBEL_3:2</a></i>;<br/><b>hence </b><a NAME="E31:30_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T8">WAYBEL_3:8</a></i>;<br/><a NAME="E32:30_3"/><i><font color="Green">E97</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font>,<font color="Maroon">T</font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">D</font>,<font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><a NAME="E33:30_3"/>
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T11">WAYBEL_3:11</a></i>;<br/><a NAME="E34:30_3"/><b>then </b>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E35:30_3"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/><a NAME="E36:30_3"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T39">WAYBEL_0:39</a></i>;<br/><a NAME="E37:30_3"/><b>then </b>
 <a href="waybel_3.html#K2">wayabove</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">D</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T12">WAYBEL_3:12</a></i>;<br/><b>hence </b><a NAME="E38:30_3"/>
<font color="Maroon">u</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:30"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">u</font></span>)</span></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> </span>}</span>  is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T32">YELLOW_9:32</a></i>;<br/>


</div>
