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<b>let </b><font color="Maroon">T</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">T</font>;<br/>



<b>thus </b><a NAME="E1:106"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">p</font>
  <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="106_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:106_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">S</font> =  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E4:106_1"/>
<font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T51">TOPS_1:51</a></i>;<br/>
<b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:106_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">u</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E7:106_1"/><i><font color="Green">E54</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/>
<a NAME="E8:106_1"/>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E9:106_1"/><b>then </b>
<font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#T5" target="_self">Th5</a>, <a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<a NAME="E10:106_1"/><b>then </b>
<font color="Maroon">p</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="waybel_3.html#T2">WAYBEL_3:2</a></i>;<br/>
<b>hence </b><a NAME="E11:106_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">p</font>
 ;<br/>


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

<a NAME="E2:106"/><i><font color="Green">E63</font></i>: 
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">p</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:106"/>
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T11">WAYBEL_3:11</a></i>;<br/>
<b>hence </b><a NAME="E4:106"/>
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T56">TOPS_1:56</a></i>;<br/>


</div>
