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

<b>thus </b><a NAME="E1:14_1_1"/>
( <font color="Maroon">X</font> is <a href="t_0topsp.html#V2">T_0</a> implies for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> or  not <font color="Olive">x</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="Olive">y</font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Olive">y</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="Olive">x</font></span><a href="domain_1.html#K6">}</a></span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">X</font> is <a href="t_0topsp.html#V2">T_0</a>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:14_1_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><b>assume </b><a NAME="E2:14_1_1_1_1"/>
( <font color="Maroon">x</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">y</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">y</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">x</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/><a NAME="E3:14_1_1_1_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</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">y</font></span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</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">x</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E4:14_1_1_1_1"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</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">y</font></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</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">x</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i/>;<br/><a NAME="E5:14_1_1_1_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</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">y</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E6:14_1_1_1_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div>
<b>end;</b></div>


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

<b>assume </b><a NAME="E2:14_1_1"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> or  not <font color="Olive">x</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="Olive">y</font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Olive">y</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="Olive">x</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>

<a NAME="E3:14_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/> <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">y</font></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:14_1_1_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/>

<b>assume </b><a NAME="E2:14_1_1_2"/><i><font color="Green">E21</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">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</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">y</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1_2_1_1"/>
(  not <font color="Maroon">x</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">y</font></span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">y</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">x</font></span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_2_1_1_1"/>
not <font color="Maroon">x</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">y</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tsp_1.html#D1" target="_self">Def1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_2_1_1_2"/>
not <font color="Maroon">y</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">x</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tsp_1.html#D1" target="_self">Def1</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:14_1_1_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:14_1_1"/>
<font color="Maroon">X</font> is <a href="t_0topsp.html#V2">T_0</a>
 <b>by </b><i/>;<br/>


</div>
