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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">P</font> be    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font>;<br/>



<a NAME="E1:62"/><i><font color="Green" title="E23">A1</font></i>: 
( <font color="Maroon" title="c2">P</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> &amp; the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">P</font> )
 
<b>by </b><i><a class="ref" href="cantor_1.html#D2" title="CANTOR_1:def.2">CANTOR_1:def 2</a></i>;<br/>
<b>let </b><font color="Maroon" title="c3">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow13.html#D4" target="_self" title="YELLOW13:def.4">YELLOW13:def 4</a><br/><b>let </b><font color="Maroon" title="c4">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow13.html#D1" target="_self" title="YELLOW13:def.1">YELLOW13:def 1</a><br/>



<b>assume </b><a NAME="E2:62"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">A</font>
 ;<br/>

<a NAME="E3:62"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">Y</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E5:62"/><i><font color="Green" title="E25">A3</font></i>: 
( <font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">Y</font> )
 <b>by </b><i><a class="txt" href="yellow13.html#E1:62"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="cantor_1.html#D1" title="CANTOR_1:def.1">CANTOR_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">Y</font> = <font color="Maroon" title="c5">Y</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>consider </b><font color="Maroon" title="c7">K</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:62"/><i><font color="Green" title="E26">A4</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">K</font> &amp; <font color="Maroon" title="c7">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> )
 <b>by </b><i><a class="txt" href="yellow13.html#E2:62"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="yellow13.html#E5:62"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">K</font> = <font color="Maroon" title="c7">K</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="yellow13.html#E8:62"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">K</font>
;<br/>

<b>thus </b><a NAME="E10:62"/>
<font color="Maroon" title="c8">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="txt" href="yellow13.html#E5:62"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="yellow13.html#E8:62"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>

<a NAME="E11:62"/><b>then </b>
<font color="Maroon" title="c8">K</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="yellow13.html#E1:62"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E12:62"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c8">K</font>
 <b>by </b><i><a class="txt" href="yellow13.html#E8:62"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/>

<a NAME="E13:62"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon" title="c8">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c6">Y</font>
 
<b>by </b><i><a class="txt" href="yellow13.html#E8:62"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E14:62"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/>
<b>hence </b><a NAME="E15:62"/>
<font color="Maroon" title="c8">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="yellow13.html#E5:62"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="yellow13.html#E13:62"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


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