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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="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:11_2_1"/><i><font color="Green">E8</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> is <a href="t_0topsp.html#V2">T_0</a>
 ;<br/>

<b>assume </b><a NAME="E2:11_2_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="tdlat_3.html#V1">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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:11_2_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>;<br/><b>thus </b><a NAME="E1:11_2_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="tsp_1.html#E2:11_2_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="tdlat_3.html#T17">TDLAT_3:17</a></i>;<br/><b>thus </b><a NAME="E2:11_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>thus </b><a NAME="E3:11_2_1_1_1"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E1:11_2_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tarski.html#D1">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">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></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">E8</font></i></a>, <a class="ref" href="tsp_1.html#D3" target="_self">Def3</a></i>;<br/>


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