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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>assume </b><a NAME="E1:21"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:21"/>
not <font color="Maroon">c<sub>1</sub></font> <a href="urysohn1.html#NR1" target="_self">is_T1</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>4</sub></font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:21"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <a href="numbers.html#K5">NAT</a>  <a href="finsop_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:21"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E6:21"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="frechet2.html#E3"><i><font color="Green">Lemma3</font></i></a></i>;<br/>
<a NAME="E7:21"/>
<font color="Maroon">c<sub>4</sub></font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E4:21"><i><font color="Green">E19</font></i></a>, <a class="ref" href="frechet.html#T23">FRECHET:23</a></i>;<br/>
<a NAME="E8:21"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="frechet.html#D4">FRECHET:def 4</a></i>;<br/>
<a NAME="E9:21"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="frechet.html#K1">Lim</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E6:21"><i><font color="Green">E21</font></i></a>, <a class="ref" href="frechet.html#D4">FRECHET:def 4</a></i>;<br/>
<b>hence </b><a NAME="E10:21"/>
contradiction
 <b>by </b><i><a class="txt" href="frechet2.html#E1:21"><i><font color="Green">E18</font></i></a>, <a class="txt" href="frechet2.html#E5:21"><i><font color="Green">E20</font></i></a>, <a class="txt" href="frechet2.html#E8:21"><i><font color="Green">E22</font></i></a></i>;<br/>


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