<?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="E1:24"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#R1">misses</a> <font color="Olive">b<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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="compts_1.html#D6">COMPTS_1:def 6</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:24"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>and </b><br/><a NAME="E3:24"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <b>and </b><br/><a NAME="E4:24"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E6:24"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E7:24"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="tietze.html#E1:24"><i><font color="Green">E18</font></i></a>, <a class="txt" href="tietze.html#E2:24"><i><font color="Green">E19</font></i></a>, <a class="txt" href="tietze.html#E3:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="tietze.html#E4:24"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E8:24"/><i><font color="Green">E24</font></i>: 
 <a href="toprealb.html#K5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="toprealb.html#D3">TOPREALB:def 3</a></i>;<br/>
<a NAME="E9:24"/><i><font color="Green">E25</font></i>: 
 <a href="toprealb.html#K5">R^1</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="toprealb.html#D3">TOPREALB:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
;<br/>

<b>thus </b><a NAME="E10:24"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>

<a NAME="E11:24"/><i><font color="Green">E26</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E12:24"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:24_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:24_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E3:24_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="tietze.html#E6:24"><i><font color="Green">E22</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:24_1"/>
0 <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T17">BORSUK_5:17</a></i>;<br/>
<b>hence </b><a NAME="E5:24_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E8:24"><i><font color="Green">E24</font></i></a>, <a class="txt" href="tietze.html#E3:24_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="tietze.html#E1:24_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="tietze.html#E11:24"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


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

<b>thus </b><a NAME="E13:24"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:24_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:24_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E3:24_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="tietze.html#E7:24"><i><font color="Green">E23</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:24_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T14">BORSUK_5:14</a></i>;<br/>
<b>hence </b><a NAME="E5:24_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E9:24"><i><font color="Green">E25</font></i></a>, <a class="txt" href="tietze.html#E3:24_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="tietze.html#E1:24_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="tietze.html#E11:24"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


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

<a NAME="E14:24"/>
 <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="tietze.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E15:24"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p2">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5">R^1</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E8:24"><i><font color="Green">E24</font></i></a>, <a class="txt" href="tietze.html#E9:24"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T141">FUNCT_1:141</a></i>;<br/>


</div>
