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<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>;<br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V3" title="TEX_2:attr.3">discrete</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">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="E1:27_1"/><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 ;<br/><a NAME="E2:27_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">G</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E4:27_1"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c5">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E5:27_1"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="tsp_1.html#E1:27"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="tex_2.html#D5" title="TEX_2:def.5">TEX_2:def 5</a></i>;<br/><b>assume </b><a NAME="E6:27_1"/><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c6">G</font> = <font color="Maroon" title="c5">G</font>;<br/><b>thus </b><a NAME="E1:27_1_1"/>
<font color="Maroon" title="c6">G</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#E4:27_1"><i><font color="Green" title="E22">A3</font></i></a></i>;<br/><a NAME="E2:27_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:27_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G</font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E5:27_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G</font>
 ;<br/><a NAME="E2:27_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="tsp_1.html#E1:27_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="tsp_1.html#E5:27_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:27_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tsp_1.html#E6:27_1"><i><font color="Green" title="E24">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="E5:27_1_1"/>
not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:27_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="c3">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="c4">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="c3">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="c4">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="E3:27"/>
<font color="Maroon" title="c2">A</font> is <a href="tsp_1.html#V1" title="TSP_1:attr.1">T_0</a>
 <b>by </b><i><a class="ref" href="tsp_1.html#D8" target="_self" title="TSP_1:def.8">Def8</a></i>;<br/>


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