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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><a NAME="E1:85"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="topgen_4.html#R1" target="_self">is_a_condensation_point_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:85"/><i><font color="Green">E47</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="topgen_4.html#R1" target="_self">is_a_condensation_point_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:85"/><i><font color="Green">E48</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="topgen_4.html#R1" target="_self">is_a_condensation_point_of</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E5:85"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> is <a href="card_4.html#V1">countable</a>
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:85"><i><font color="Green">E47</font></i></a>, <a class="ref" href="topgen_4.html#D9" target="_self">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E7:85"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> is <a href="card_4.html#V1">countable</a>
 <b>by </b><i><a class="txt" href="topgen_4.html#E3:85"><i><font color="Green">E48</font></i></a>, <a class="ref" href="topgen_4.html#D9" target="_self">Def9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="connsp_2.html#T4">CONNSP_2:4</a></i>;<br/>
<a NAME="E9:85"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E10:85"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E11:85"/>
<font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_4.html#E9:85"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E12:85"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> is <a href="card_4.html#V1">countable</a>
 
<b>by </b><i><a class="ref" href="card_4.html#T46">CARD_4:46</a>, <a class="txt" href="topgen_4.html#E5:85"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E13:85"/>
<font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_4.html#E10:85"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E14:85"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> is <a href="card_4.html#V1">countable</a>
 
<b>by </b><i><a class="ref" href="card_4.html#T46">CARD_4:46</a>, <a class="txt" href="topgen_4.html#E7:85"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E15:85"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="card_4.html#V1">countable</a>
 
<b>by </b><i><a class="txt" href="topgen_4.html#E12:85"><i><font color="Green">E53</font></i></a>, <a class="ref" href="card_4.html#T47">CARD_4:47</a></i>;<br/>
<a NAME="E16:85"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="card_4.html#V1">countable</a>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<b>hence </b><a NAME="E17:85"/>
contradiction
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:85"><i><font color="Green">E46</font></i></a>, <a class="ref" href="topgen_4.html#D9" target="_self">Def9</a></i>;<br/>


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