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<b>let </b><font color="Maroon">L</font> be  <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">X</font> be  <a href="waybel_0.html#V13">upper</a> <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="E1:14_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">X</font> is <a href="waybel_6.html#V1" target="_self">Open</a>
 ;<br/><a NAME="E2:14_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <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">L</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_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="E1:14_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:14_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x1</font> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E6:14_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y1</font> where <font color="Olive">y1</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y1</font> <a href="waybel_3.html#NR3" target="_self">&gt;&gt;</a> <font color="Maroon">y</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E7:14_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D4">WAYBEL_3:def 4</a></i>;<br/>
<a NAME="E8:14_1_1"/>
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">y</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">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:14_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <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">L</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:14_1"/>
 <a href="tarski.html#K3">union</a> <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">L</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_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="E1:14_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <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">L</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:14_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E4:14_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">Y</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">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:14_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:14_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</font> <a href="waybel_3.html#NR3" target="_self">&gt;&gt;</a> <font color="Maroon">x</font> </span>}</span> 
 
<b>by </b><i>, , <a class="ref" href="waybel_3.html#D4">WAYBEL_3:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:14_1_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z1</font> <a href="waybel_3.html#NR3" target="_self">&gt;&gt;</a> <font color="Maroon">x</font> )
 ;<br/>
<a NAME="E10:14_1_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z1</font>
 
<b>by </b><i>, <a class="ref" href="waybel_3.html#T1">WAYBEL_3:1</a></i>;<br/>
<b>hence </b><a NAME="E11:14_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</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="E4:14_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <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">L</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:14"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <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">L</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:14_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:14_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E4:14_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">Y</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">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:14_2"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/><a NAME="E7:14_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</font> <a href="waybel_3.html#NR3" target="_self">&gt;&gt;</a> <font color="Maroon">x</font> </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="waybel_3.html#D4">WAYBEL_3:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:14_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">z1</font> <a href="waybel_3.html#NR3" target="_self">&gt;&gt;</a> <font color="Maroon">x</font> )
 ;<br/><b>thus </b><a NAME="E10:14_2"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x1</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:14"/>
<font color="Maroon">X</font> is <a href="waybel_6.html#V1" target="_self">Open</a>
 <b>by </b><i/>;<br/>


</div>
