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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b3">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  holds <br/>(  not <font color="Olive" title="b3">x</font> <a href="topgen_4.html#R1" title="TOPGEN_4:pred.1">is_a_condensation_point_of</a> <font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font> or <font color="Olive" title="b3">x</font> <a href="topgen_4.html#R1" title="TOPGEN_4:pred.1">is_a_condensation_point_of</a> <font color="Olive" title="b1">A</font> or <font color="Olive" title="b3">x</font> <a href="topgen_4.html#R1" title="TOPGEN_4:pred.1">is_a_condensation_point_of</a> <font color="Olive" title="b2">B</font> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  holds <br/>(  not <font color="Olive" title="b1">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> or <font color="Olive" title="b1">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="Olive" title="b1">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> )</span><br/><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">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> or <font color="Maroon" title="c4">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="c4">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> )</span><br/>







<b>assume </b><a NAME="E1:84"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c4">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">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="c4">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> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:84"/><i><font color="Green" title="E47">A2</font></i>: 
not <font color="Maroon" title="c4">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>
 <b>and </b><br/><a NAME="E3:84"/><i><font color="Green" title="E48">A3</font></i>: 
not <font color="Maroon" title="c4">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c5">N1</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">x</font><b> such that </b><br/><a NAME="E5:84"/><i><font color="Green" title="E49">A4</font></i>: 
<font color="Maroon" title="c5">N1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">A</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:84"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="topgen_4.html#D9" target="_self" title="TOPGEN_4:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">N2</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">x</font><b> such that </b><br/><a NAME="E7:84"/><i><font color="Green" title="E50">A5</font></i>: 
<font color="Maroon" title="c6">N2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 <b>by </b><i><a class="txt" href="topgen_4.html#E3:84"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="topgen_4.html#D9" target="_self" title="TOPGEN_4:def.9">Def9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">N3</font> = <font color="Maroon" title="c5">N1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">N2</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">x</font> <b>by </b><i><a class="ref" href="connsp_2.html#T4" title="CONNSP_2:th.4">CONNSP_2:4</a></i>;<br/>
<a NAME="E9:84"/><i><font color="Green" title="E51">A6</font></i>: 
<font color="Maroon" title="c7">N3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">N1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E10:84"/><i><font color="Green" title="E52">A7</font></i>: 
<font color="Maroon" title="c7">N3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">N2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E11:84"/><i><font color="Green" title="E53">A8</font></i>: 
<font color="Maroon" title="c7">N3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">A</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 
<b>by </b><i><a class="txt" href="topgen_4.html#E5:84"><i><font color="Green" title="E49">A4</font></i></a>, <a class="txt" href="topgen_4.html#E9:84"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="card_4.html#T46" title="CARD_4:th.46">CARD_4:46</a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E12:84"/>
<font color="Maroon" title="c7">N3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 
<b>by </b><i><a class="txt" href="topgen_4.html#E7:84"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="topgen_4.html#E10:84"><i><font color="Green" title="E52">A7</font></i></a>, <a class="ref" href="card_4.html#T46" title="CARD_4:th.46">CARD_4:46</a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E13:84"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">N3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">N3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 
<b>by </b><i><a class="txt" href="topgen_4.html#E11:84"><i><font color="Green" title="E53">A8</font></i></a>, <a class="ref" href="card_4.html#T47" title="CARD_4:th.47">CARD_4:47</a></i>;<br/>
<a NAME="E14:84"/><b>then </b>
<font color="Maroon" title="c7">N3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>;<br/>
<b>hence </b><a NAME="E15:84"/>
contradiction
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:84"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="topgen_4.html#D9" target="_self" title="TOPGEN_4:def.9">Def9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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