<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">T</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="waybel_9.html#L1" title="WAYBEL_9:struct.1">TopRelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  &amp; <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V2" title="PRE_TOPC:attr.2">TopSpace-like</a>  implies <font color="Maroon" title="c1">T</font> is  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_1_1"/>
( <font color="Maroon" title="c1">T</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  &amp; <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V2" title="PRE_TOPC:attr.2">TopSpace-like</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">T</font> is  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">W</font> = <font color="Maroon" title="c1">T</font> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="pre_topc.html#V2" title="PRE_TOPC:attr.2">TopSpace-like</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="waybel_9.html#L1" title="WAYBEL_9:struct.1">TopRelStr</a>  ;<br/>
<a NAME="E3:21_1_1"/>
<font color="Maroon" title="c2">W</font> is  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">W</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel11.html#D5" title="WAYBEL11:def.5">WAYBEL11:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  not <font color="Maroon" title="c3">S</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  or <font color="Maroon" title="c3">S</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>  ) &amp; (  not <font color="Maroon" title="c3">S</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>  or <font color="Maroon" title="c3">S</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:21_1_1_1"/>
( <font color="Maroon" title="c3">S</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies <font color="Maroon" title="c3">S</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">S</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>  or <font color="Maroon" title="c3">S</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:21_1_1_1"/>
(  not <font color="Maroon" title="c3">S</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>  or <font color="Maroon" title="c3">S</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tdlat_3.html#T17" title="TDLAT_3:th.17">TDLAT_3:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:21_1_1"/>
<font color="Maroon" title="c1">T</font> is  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
