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<b>let </b><font color="Maroon">N</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_2.html#V2">meet-continuous</a> <a href="waybel19.html#V2">Lawson</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">S</font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">N</font>;<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">N</font>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>







<b>assume </b><a NAME="E1:31"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:31"/><i><font color="Green">E34</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E3:31"/><i><font color="Green">E35</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T16">WAYBEL_0:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">sX</font> =  <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">K</font> =  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E6:31"/>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">sX</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<a NAME="E7:31"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">K</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T13">WAYBEL_0:13</a></i>;<br/>
<a NAME="E8:31"/>
<font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:31_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E4:31_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/>
<a NAME="E5:31_1"/><i><font color="Green">E45</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E6:31_1"/>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_1_1"/>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T24">WAYBEL_0:24</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:31_1_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T16">WAYBEL_0:16</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:31_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , , <a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:31"/><b>then </b>
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T56">TOPS_1:56</a></i>;<br/>
<b>hence </b><a NAME="E10:31"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>reconsider </b><font color="Maroon">A</font> =  <a href="tops_1.html#K1">Int</a> <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<a NAME="E12:31"/>
<font color="Maroon">N</font> is  <a href="pre_topc.html#V2">correct</a> <a href="waybel19.html#V2">Lawson</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E13:31"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="waybel19.html#T37">WAYBEL19:37</a></i>;<br/>
<a NAME="E14:31"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<b>hence </b><a NAME="E15:31"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T56">TOPS_1:56</a></i>;<br/>


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