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<b>let </b><font color="Maroon">S</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">T</font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1" target="_self">TopAugmentation</a> of <font color="Maroon">S</font>;<br/>



<b>set </b><font color="Maroon">R</font> =  <a href="waybel_9.html#G1">TopRelStr</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>,<span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font></span>)</span> #);<br/>
<a NAME="E1:78"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <a href="waybel_9.html#G1">TopRelStr</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>,<span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font></span>)</span> #),the <a href="orders_2.html#U1">InternalRel</a> of <a href="waybel_9.html#G1">TopRelStr</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>,<span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font></span>)</span> #) #) <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> #)
 
;<br/>
<b>then reconsider </b><font color="Maroon">R</font> =  <a href="waybel_9.html#G1">TopRelStr</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>,<span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font></span>)</span> #) as    <a href="yellow_9.html#M1" target="_self">TopAugmentation</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">R</font> = <font color="Maroon">R</font> as  <a href="pre_topc.html#V2">correct</a> <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1" target="_self">TopAugmentation</a> of <font color="Maroon">S</font> <b>by </b><i>, </i>;<br/>
<a NAME="E4:78"/><i><font color="Green">E32</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</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">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> #)
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:78"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</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="78_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:78_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E4:78_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:78_1"/><b>then </b>
<font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="txt" href="yellow_9.html#E6:78"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:78_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</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>

<b>let </b><font color="Maroon">x</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="E6:78"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> ;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E9:78"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E10:78"/><b>then </b>
<font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="txt" href="yellow_9.html#E6:78"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:78"/>
<font color="Maroon">x</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/>


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