<?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="E1:2"/>
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> 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="E3:2"/><i><font color="Green">E3</font></i>: 
 <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E4:2"/>
(  not <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="tarski.html#R1">c=</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> or  not  <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> <a href="tarski.html#R1">c=</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#E3:2"><i><font color="Green">E3</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:2"/><i><font color="Green">E4</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>and </b><br/><a NAME="E7:2"/><i><font color="Green">E5</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</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="ref" href="pre_topc.html#T48">PRE_TOPC:48</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="frechet2.html#E6:2"><i><font color="Green">E4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E9:2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="frechet2.html#E7:2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>

<b>thus </b><a NAME="E10:2"/>
<font color="Maroon">c<sub>4</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#E6:2"><i><font color="Green">E4</font></i></a></i>;<br/>


</div>
