<?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="E1:30"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</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><br/>for <font color="Olive">b<sub>2</sub></font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span>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>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> st <font color="Olive">b<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>

<a NAME="E2:30"/>
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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  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>;<br/>

<b>assume </b><a NAME="E1:30_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<a NAME="E2:30_1"/><i><font color="Green">E24</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="pre_topc.html#V5">continuous</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="pre_topc.html#V5">continuous</a> )
 
<b>by </b><i><a class="ref" href="tmap_1.html#T68">TMAP_1:68</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tmap_1.html#K10">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E3:30_1"/><i><font color="Green">E25</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E4:30_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E5:30_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="tietze.html#E3:30_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tsep_1.html#D2">TSEP_1:def 2</a></i>;<br/>
<a NAME="E6:30_1"/><i><font color="Green">E27</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E7:30_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="tietze.html#E1:30_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="tietze.html#E3:30_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tsep_1.html#D3">TSEP_1:def 3</a></i>;<br/>
<a NAME="E8:30_1"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p4">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tmap_1.html#K10">union</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p4">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p4">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p4">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tietze.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E9:30_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="tmap_1.html#D3">TMAP_1:def 3</a></i>;<br/>
<a NAME="E10:30_1"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E11:30_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p4">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p4">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E12:30_1"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p4">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tmap_1.html#K10">union</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p4">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="tietze.html#E8:30_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<i><font color="Green">E31</font></i>: <font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span>

<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K1">--&gt;</a> 0
<b>by </b><i><a class="ref" href="toprealb.html#D2">TOPREALB:def 2</a></i>
;<br/>
<a NAME="E14:30_1"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <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>by </b><i><a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/>
<i><font color="Green">E33</font></i>: <font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K1">--&gt;</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span>

<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K1">--&gt;</a> 1
<b>by </b><i><a class="ref" href="toprealb.html#D2">TOPREALB:def 2</a></i>
;<br/>
<a NAME="E16:30_1"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <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="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/>
<a NAME="E17:30_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p4">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tmap_1.html#K10">union</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p4">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</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="E1:30_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p4">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tmap_1.html#K10">union</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p4">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:30_1_3"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</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#E14:30_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="tietze.html#E16:30_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="tietze.html#E12:30_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:30_1_3"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:30_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tmap_1.html#K10">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="txt" href="tietze.html#E5:30_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="tietze.html#E6:30_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_2.html#T8">FUNCT_2:8</a></i>;<br/>
<a NAME="E19:30_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tmap_1.html#K10">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="txt" href="tietze.html#E2:30_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="tietze.html#E7:30_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tmap_1.html#T148">TMAP_1:148</a></i>;<br/>
<a NAME="E20:30_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="tietze.html#E5:30_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="topmetr.html#T8">TOPMETR:8</a></i>;<br/>
<a NAME="E21:30_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T36">TOPS_1:36</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</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>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span><b> such that </b><br/><a NAME="E23:30_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tmap_1.html#K10">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="tietze.html#E1:30"><i><font color="Green">E22</font></i></a>, <a class="txt" href="tietze.html#E20:30_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as  <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>by </b><i><a class="ref" href="topmetr.html#T7">TOPMETR:7</a>, <a class="ref" href="topreala.html#T28">TOPREALA:28</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<a NAME="E25:30_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E26:30_1"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:30_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="tietze.html#E13:30_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="borsuk_1.html#T6">BORSUK_1:6</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>12</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="E2:30_1_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:30_1_4"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:30_1_4"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <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="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:30_1_4"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E7:30_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="tietze.html#E13:30_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="tietze.html#E6:30_1_4"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E8:30_1_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="tietze.html#E6:30_1_4"><i><font color="Green">E41</font></i></a>, <a class="txt" href="tietze.html#E4:30_1_4"><i><font color="Green">E40</font></i></a>, <a class="txt" href="tietze.html#E3:30_1_4"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:30_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p4">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tmap_1.html#K10">union</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p4">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="tietze.html#E23:30_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tmap_1.html#T4">TMAP_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="tietze.html#E7:30_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="tietze.html#E25:30_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="tietze.html#E3:30_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tietze.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="tietze.html#E3:30_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tmap_1.html#D3">TMAP_1:def 3</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="tietze.html#E26:30_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="extens_1.html#T1">EXTENS_1:1</a></i>
;<br/><br/>

<a NAME="E29:30_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E30:30_1"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:30_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</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#E15:30_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="borsuk_1.html#T6">BORSUK_1:6</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>12</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="E2:30_1_6"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:30_1_6"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:30_1_6"/><i><font color="Green">E42</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <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="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:30_1_6"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E7:30_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="tietze.html#E15:30_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="tietze.html#E6:30_1_6"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E8:30_1_6"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="tietze.html#E6:30_1_6"><i><font color="Green">E43</font></i></a>, <a class="txt" href="tietze.html#E4:30_1_6"><i><font color="Green">E42</font></i></a>, <a class="txt" href="tietze.html#E3:30_1_6"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:30_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p4">(<span class="default"><a href="toprealb.html#K4">R^1</a> 0</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tmap_1.html#K10">union</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p4">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="tietze.html#E23:30_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tmap_1.html#T4">TMAP_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="tmap_1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="tietze.html#E7:30_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="tietze.html#E29:30_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="tietze.html#E3:30_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tietze.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K3">--&gt;</a> <span class="p3">(<span class="default"><a href="toprealb.html#K4">R^1</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="tietze.html#E3:30_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tmap_1.html#D3">TMAP_1:def 3</a></i>
<br/>.= 
<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#E30:30_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="extens_1.html#T1">EXTENS_1:1</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:30"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="compts_1.html#V5">being_T4</a>
 <b>by </b><i><a class="ref" href="tietze.html#T20" target="_self">Th20</a></i>;<br/>


</div>
