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<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:11_2_1"/><i><font color="Green" title="E8">A3</font></i>: 
not <font color="Maroon" title="c1">Y</font> is <a href="t_0topsp.html#V2" title="T_0TOPSP:attr.2">T_0</a>
 ;<br/>

<b>assume </b><a NAME="E2:11_2_1"/><i><font color="Green" title="E9">A4</font></i>: 
<font color="Maroon" title="c1">Y</font> is <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font>;<br/><b>assume </b><a NAME="E1:11_2_1_1"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c4">V</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><b>thus </b><a NAME="E1:11_2_1_1_1"/>
<font color="Maroon" title="c4">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="tsp_1.html#E2:11_2_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="tdlat_3.html#T17" title="TDLAT_3:th.17">TDLAT_3:17</a></i>;<br/><b>thus </b><a NAME="E2:11_2_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>thus </b><a NAME="E3:11_2_1_1_1"/>
not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E1:11_2_1_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:11_2_1_1"/>
(  ex <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> ) or  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> ) )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:11_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tsp_1.html#E1:11_2_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="tsp_1.html#D3" target="_self" title="TSP_1:def.3">Def3</a></i>;<br/>


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