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

<b>thus </b><a NAME="E1:7_1_1"/>
(  not <font color="Maroon">c<sub>1</sub></font> is <a href="t_0topsp.html#V2">T_0</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> or for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> or ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> ) or ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_1_1"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> or for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> or ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> ) or ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> ) ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="tsp_1.html#D3" target="_self">TSP_1:def 3</a><br/>

<b>assume </b><a NAME="E2:7_1_1_1"/><i><font color="Green">E7</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E3:7_1_1_1"/><i><font color="Green">E8</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="tsp_1.html#E2:7_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E4:7_1_1_1"/><i><font color="Green">E9</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>
 ;<br/>

<div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_1_1_1_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="tsp_1.html#E1:7_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="tsp_1.html#E2:7_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="tsp_1.html#E4:7_1_1_1"><i><font color="Green">E9</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_1_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:7_1_1_1_1_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:7_1_1_1_1_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E5:7_1_1_1_1_1_1"/><i><font color="Green">E12</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> ;<br/><a NAME="E1:7_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/><b>hence </b><a NAME="E2:7_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="tsp_1.html#E3:7_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/><b>thus </b><a NAME="E3:7_1_1_1_1_1_1_1"/>
not <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="txt" href="tsp_1.html#E4:7_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>thus </b><a NAME="E4:7_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</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="tsp_1.html#E3:7_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="tsp_1.html#E5:7_1_1_1_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:7_1_1_1_1_1_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_1_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:7_1_1_1_1_1_2"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:7_1_1_1_1_1_2"/><i><font color="Green">E11</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E5:7_1_1_1_1_1_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> ;<br/><a NAME="E1:7_1_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/><b>hence </b><a NAME="E2:7_1_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="tsp_1.html#E3:7_1_1_1_1_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/><b>thus </b><a NAME="E3:7_1_1_1_1_1_2_1"/>
<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="txt" href="tsp_1.html#E3:7_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="tsp_1.html#E4:7_1_1_1_1_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/><b>thus </b><a NAME="E4:7_1_1_1_1_1_2_1"/>
not <font color="Maroon">c<sub>3</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="tsp_1.html#E5:7_1_1_1_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:7_1_1_1_1_1_2"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>


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

<b>assume </b><a NAME="E2:7_1_1"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> or for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> or ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> ) or ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> ) ) )
 ;<br/>

<a NAME="E3:7_1_1"/>
( not <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> implies for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> or ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> ) or ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_1_2"/><i><font color="Green">E7</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E2:7_1_1_2"/><i><font color="Green">E8</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="tsp_1.html#E1:7_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E3:7_1_1_2"/><i><font color="Green">E9</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>
 ;<br/>

<div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_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="E1:7_1_1_2_1_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="tsp_1.html#E2:7_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="tsp_1.html#E1:7_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="tsp_1.html#E3:7_1_1_2"><i><font color="Green">E9</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_1_2_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:7_1_1_2_1_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E4:7_1_1_2_1_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E5:7_1_1_2_1_1_1"/><i><font color="Green">E12</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> ;<br/><a NAME="E1:7_1_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>hence </b><a NAME="E2:7_1_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="tsp_1.html#E3:7_1_1_2_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/><b>thus </b><a NAME="E3:7_1_1_2_1_1_1_1"/>
not <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="txt" href="tsp_1.html#E4:7_1_1_2_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>thus </b><a NAME="E4:7_1_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>3</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="tsp_1.html#E2:7_1_1_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="tsp_1.html#E5:7_1_1_2_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:7_1_1_2_1_1_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_1_2_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:7_1_1_2_1_1_2"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E4:7_1_1_2_1_1_2"/><i><font color="Green">E11</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E5:7_1_1_2_1_1_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> ;<br/><a NAME="E1:7_1_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>hence </b><a NAME="E2:7_1_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="tsp_1.html#E3:7_1_1_2_1_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/><b>thus </b><a NAME="E3:7_1_1_2_1_1_2_1"/>
<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="txt" href="tsp_1.html#E2:7_1_1_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="tsp_1.html#E4:7_1_1_2_1_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/><b>thus </b><a NAME="E4:7_1_1_2_1_1_2_1"/>
not <font color="Maroon">c<sub>3</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="tsp_1.html#E5:7_1_1_2_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:7_1_1_2_1_1_2"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:7_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="t_0topsp.html#V2">T_0</a>
 <b>by </b><i><a class="ref" href="tsp_1.html#D3" target="_self">Def3</a></i>;<br/>


</div>
