<?xml version="1.0"?>
<div class="add">

<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="E1:3"/>
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><b> such that </b><br/><a NAME="E3:3"/><i><font color="Green">E4</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="E4:3"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="frechet2.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <a href="numbers.html#K5">NAT</a>  <a href="finsop_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>take </b>
<a href="numbers.html#K5">NAT</a>  <a href="finsop_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>thus </b><a NAME="E5:3"/>
<a href="numbers.html#K5">NAT</a>  <a href="finsop_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>2</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>
 ;<br/>

<b>thus </b><a NAME="E6:3"/>
<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>by </b><i><a class="txt" href="frechet2.html#E3:3"><i><font color="Green">E4</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:3"/>
<a href="numbers.html#K5">NAT</a>  <a href="finsop_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="frechet.html#D2">FRECHET:def 2</a><br/>

<b>assume </b><a NAME="E1:3_1"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E2:3_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E4:3"><i><font color="Green">E5</font></i></a>, <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/>
<a NAME="E3:3_1"/><b>then </b><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/>
<b>take </b>
0
;<br/>

<b>thus </b><a NAME="E4:3_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/>(  not 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="finsop_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="frechet2.html#E3:3_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


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


</div>
