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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>







<span class="kw">assume </span><a NAME="E1:85"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:85"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
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>
 <span class="kw">and </span><br/><a NAME="E3:85"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:85"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
<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_3.html#V4" title="CARD_3:attr.4">countable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:85"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="topgen_4.html#D9" target="_self" title="TOPGEN_4:def.9">Def9</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E7:85"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
<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_3.html#V4" title="CARD_3:attr.4">countable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="topgen_4.html#D9" target="_self" title="TOPGEN_4:def.9">Def9</a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#T4" title="CONNSP_2:th.4">CONNSP_2:4</a></span>;<br/>
<a NAME="E9:85"/>
( <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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> &amp; <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></span>;<br/>
<a NAME="E10:85"/><span class="kw">then </span>
<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_3.html#V4" title="CARD_3:attr.4">countable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:85"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E7:85"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="card_3.html#T128" title="CARD_3:th.128">CARD_3:128</a></span>;<br/>
<a NAME="E11:85"/><span class="kw">then </span>
<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_3.html#V4" title="CARD_3:attr.4">countable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:85"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="topgen_4.html#D9" target="_self" title="TOPGEN_4:def.9">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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