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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:6_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">T</font> is <a href="t_0topsp.html#V2">T_0</a>
 ;<br/><b>assume </b><a NAME="E2:6_1_1_1"/><i><font color="Green">E18</font></i>: 
not <font color="Maroon">T</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E3:6_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E4:6_1_1_1"/><b>then </b>
 ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; ( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) or ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) ) )
 <b>by </b><i>, , <a class="ref" href="t_0topsp.html#D7">T_0TOPSP:def 7</a></i>;<br/><b>hence </b><a NAME="E5:6_1_1_1"/>
(  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) or  ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> ) )
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:6_1_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">T</font> is <a href="struct_0.html#V3">empty</a> or for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</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  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) or  ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> ) ) )
 ;<br/>

<b>assume </b><a NAME="E3:6_1_1"/><i><font color="Green">E20</font></i>: 
not <font color="Maroon">T</font> is <a href="struct_0.html#V3">empty</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D7">T_0TOPSP:def 7</a><br/>

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

<b>assume </b><a NAME="E4:6_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E5:6_1_1"/><b>then </b>
(  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) or  ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:6_1_1"/>
 ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; ( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) or ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) ) )
 ;<br/>


</div>
