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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="waybel_9.html#L1">TopRelStr</a> ;<br/>

<b>assume </b><a NAME="E1:21_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="realset2.html#V3">trivial</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="pre_topc.html#V2">TopSpace-like</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V2">TopSpace-like</a> <a href="realset2.html#V3">trivial</a> <a href="orders_2.html#V2">reflexive</a>  <a href="waybel_9.html#L1">TopRelStr</a>  ;<br/>
<a NAME="E3:21_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel11.html#V4">Scott</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D5">WAYBEL11:def 5</a><br/>

<b>thus </b><a NAME="E1:21_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> implies <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V13">upper</a> )
 ;<br/>

<b>thus </b><a NAME="E2:21_1_1_1"/>
(  not <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V13">upper</a> or <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="tdlat_3.html#T17">TDLAT_3:17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:21_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel11.html#V4">Scott</a>
 ;<br/>


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