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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</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>thus </b><a NAME="E1:85"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a> </span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:85_1"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E3:85_1"/>
<font color="Maroon" title="c5">x'</font> <a href="topgen_4.html#R1" title="TOPGEN_4:pred.1">is_a_condensation_point_of</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="txt" href="topgen_4.html#E1:85_1"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="topgen_4.html#D10" target="_self" title="TOPGEN_4:def.10">Def10</a></i>;<br/>
<a NAME="E4:85_1"/><b>then </b>
( <font color="Maroon" title="c5">x'</font> <a href="topgen_4.html#R1" title="TOPGEN_4:pred.1">is_a_condensation_point_of</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c5">x'</font> <a href="topgen_4.html#R1" title="TOPGEN_4:pred.1">is_a_condensation_point_of</a> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="ref" href="topgen_4.html#T56" target="_self" title="TOPGEN_4:th.56">Th56</a></i>;<br/>
<a NAME="E5:85_1"/><b>then </b>
( <font color="Maroon" title="c5">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a>  or <font color="Maroon" title="c5">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a>  )
 
<b>by </b><i><a class="ref" href="topgen_4.html#D10" target="_self" title="TOPGEN_4:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E6:85_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E2:85"/><i><font color="Green" title="E46">A2</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a> 
 
<b>by </b><i><a class="ref" href="topgen_4.html#T55" target="_self" title="TOPGEN_4:th.55">Th55</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:85"/>
<font color="Maroon" title="c3">B</font> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a> 
 
<b>by </b><i><a class="ref" href="topgen_4.html#T55" target="_self" title="TOPGEN_4:th.55">Th55</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E4:85"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="topgen_4.html#K4" title="TOPGEN_4:func.4">^0</a> 
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:85"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>


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