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<b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/>

<b>assume </b><a NAME="E1:8_1_1"/>
<font color="Maroon">Y</font> is <a href="realset2.html#V3">trivial</a>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:8_1_1"/><i><font color="Green">E17</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tex_1.html#D1">TEX_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>;<br/><a NAME="E1:8_1_1_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>assume </b><a NAME="E2:8_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><b>hence </b><a NAME="E3:8_1_1_1"/>
(  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) or  ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <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="E5:8_1_1"/>
<font color="Maroon">Y</font> is <a href="t_0topsp.html#V2">T_0</a>
 <b>by </b><i/>;<br/>


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