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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="realset2.html#V3">trivial</a> <a href="tdlat_3.html#V2">anti-discrete</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:9_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/><b>take </b><font color="Maroon">x</font> = <font color="Maroon">x</font>;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><b>thus </b><a NAME="E3:9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><div><i><font color="Green">E18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:9_1_1"/>
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E2:9_1_1"/><b>then </b><i><font color="Green">E19</font></i>: 
( <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="tdlat_3.html#T20">TDLAT_3:20</a></i>;<br/><b>assume </b><a NAME="E3:9_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/><b>hence </b><a NAME="E4:9_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:9_1_2"/>
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E2:9_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
( <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="tdlat_3.html#T20">TDLAT_3:20</a></i>;<br/><b>assume </b><a NAME="E3:9_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/><b>hence </b><a NAME="E4:9_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:9_1"/>
( ( for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  holds <br/>(  not <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) ) &amp; ( for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  holds <br/>(  not <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> ) ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:9"/>
not <font color="Maroon">X</font> is <a href="t_0topsp.html#V2">T_0</a>
 <b>by </b><i/>;<br/>


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