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<b>let </b><font color="Maroon" title="c1">T</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">U</font>, <font color="Maroon" title="c3">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c4">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:88"/><i><font color="Green" title="E67">A1</font></i>: 
( <font color="Maroon" title="c2">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c3">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c4">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font> )
 ;<br/>

<a NAME="E2:88"/><i><font color="Green" title="E68">A2</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">B'</font> = <font color="Maroon" title="c4">B</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="topgen_5.html#E1:88"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:88"/>
<font color="Maroon" title="c4">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">V</font></span>)</span></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> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="topgen_5.html#E1:88"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="cantor_1.html#D2" title="CANTOR_1:def.2">CANTOR_1:def 2</a></i>;<br/>
<a NAME="E5:88"/><b>then </b><i><font color="Green" title="E69">A3</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="topgen_5.html#E2:88"><i><font color="Green" title="E68">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/><b>assume </b><a NAME="E1:88_1"/><i><font color="Green" title="E70">A4</font></i>: 
<font color="Maroon" title="c6">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/><b>let </b><font color="Maroon" title="c7">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>;<br/><b>assume </b><a NAME="E2:88_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">A'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:88_1"/><i><font color="Green" title="E71">A5</font></i>: 
( <font color="Maroon" title="c8">A'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">U</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A'</font> &amp; <font color="Maroon" title="c8">A'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:88"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="topgen_5.html#E1:88_1"><i><font color="Green" title="E70">A4</font></i></a>, <a class="ref" href="yellow_9.html#T31" title="YELLOW_9:th.31">YELLOW_9:31</a></i>;<br/><b>assume </b><a NAME="E5:88_1"/><i><font color="Green" title="E72">A6</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  holds <br/>(  not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B'</font> or  not <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">a</font> or  not <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> )
 ;<br/><a NAME="E6:88_1"/><i><font color="Green" title="E73">A7</font></i>: 
( <font color="Maroon" title="c2">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">U</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">U</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">V</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E7:88_1"/>
( <font color="Maroon" title="c8">A'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> or <font color="Maroon" title="c8">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">V</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E4:88_1"><i><font color="Green" title="E71">A5</font></i></a>, <a class="ref" href="setwiseo.html#T6" title="SETWISEO:th.6">SETWISEO:6</a></i>;<br/><a NAME="E8:88_1"/><b>then </b>
( ( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font> &amp; <font color="Maroon" title="c2">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> ) or ( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">A</font> ) )
 <b>by </b><i><a class="txt" href="topgen_5.html#E4:88_1"><i><font color="Green" title="E71">A5</font></i></a>, <a class="txt" href="topgen_5.html#E5:88_1"><i><font color="Green" title="E72">A6</font></i></a>, <a class="txt" href="topgen_5.html#E6:88_1"><i><font color="Green" title="E73">A7</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E9:88_1"/>
contradiction
 <b>by </b><i><a class="txt" href="topgen_5.html#E1:88"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="topgen_5.html#E5:88_1"><i><font color="Green" title="E72">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:88"/>
<font color="Maroon" title="c4">B</font> is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="topgen_5.html#E5:88"><i><font color="Green" title="E69">A3</font></i></a>, <a class="ref" href="yellow_9.html#T32" title="YELLOW_9:th.32">YELLOW_9:32</a></i>;<br/>


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