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

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">Ff</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>







<b>assume </b><a NAME="E1:9"/><i><font color="Green">E24</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a>  &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> &amp; <font color="Maroon">Ff</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">Ff</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">Ff</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> )
 ;<br/>

<a NAME="E2:9"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E3:9"/><i><font color="Green">E33</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> 0,<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> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; 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">1 <a href="real_1.html#K6">/</a> 2</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">1 <a href="real_1.html#K6">/</a> 2</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="E4:9"/><b>then </b><i><font color="Green">E36</font></i>: 
(  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<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> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp;  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</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">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 
;<br/>
<a NAME="E5:9"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <a href="zfmisc_1.html#K1">K18</a>(the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>) =  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> implies  ex <font color="Olive">NE</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">NE</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">F</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Olive">NE</font></span>)</span>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Olive">F</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> holds <br/> ex <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">p1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p1</font> &amp; <font color="Olive">p1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> &amp; <font color="Olive">p2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">p1</font>,<font color="Olive">p2</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font> &amp; <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) ) ) );<br/>
<b>reconsider </b><font color="Maroon">h1</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<i><font color="Green">E38</font></i>: <font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> = 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>
;<br/>
<a NAME="E8:9"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_2"/><i><font color="Green">E52</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> 2 &amp; 0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E2:9_2"/><b>then </b><i><font color="Green">E53</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">NE</font> = <font color="Maroon">H<sub>1</sub></font>(0) as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal1.html#T29">TOPREAL1:29</a></i>;<br/>
<b>take </b>
<font color="Maroon">NE</font>
;<br/>

<b>thus </b><a NAME="E4:9_2"/>
<font color="Maroon">NE</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(0)
 ;<br/>

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE</font></span>)</span>;<br/>



<b>assume </b><a NAME="E5:9_2"/><i><font color="Green">E54</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Maroon">F</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span> )
 ;<br/>

<a NAME="E6:9_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E7:9_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E8:9_2"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E9:9_2"/><i><font color="Green">E57</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:9_2"/><b>then </b><i><font color="Green">E58</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> <a href="funct_1.html#K1">.</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> <a href="funct_1.html#K1">.</a> 2 )
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E11:9_2"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/>
<a NAME="E12:9_2"/><i><font color="Green">E60</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E61</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE</font></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
 <a href="topreal1.html#K5">L~</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="sppol_2.html#T21">SPPOL_2:21</a></i>
<br/>.= 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,1
<b>by </b><i>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
;<br/>
<b>take </b>
0
;<br/>

<b>take </b>
1
;<br/>

<b>thus </b><a NAME="E14:9_2"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#T45">FUNCT_2:45</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:9"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:9_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>assume </b><a NAME="E2:9_3"/><i><font color="Green">E65</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E3:9_3"/><i><font color="Green">E66</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:9_3"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>then consider </b><font color="Maroon">NE</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E6:9_3"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">NE</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font>) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">F</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE</font></span>)</span>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Olive">F</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> holds <br/> ex <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">p1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p1</font> &amp; <font color="Olive">p1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> &amp; <font color="Olive">p2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">p1</font>,<font color="Olive">p2</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font> &amp; <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) ) )
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:9_3"/><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E8:9_3"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E9:9_3"/><i><font color="Green">E70</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E10:9_3"/><b>then </b><i><font color="Green">E71</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:9_3"/><i><font color="Green">E72</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:9_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E13:9_3"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/>
<a NAME="E14:9_3"/><i><font color="Green">E75</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">NE1</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1) as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal1.html#T29">TOPREAL1:29</a></i>;<br/>
<b>take </b>
<font color="Maroon">NE1</font>
;<br/>

<b>thus </b><a NAME="E16:9_3"/>
<font color="Maroon">NE1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1)
 ;<br/>

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE1</font></span>)</span>;<br/>



<b>assume </b><a NAME="E17:9_3"/><i><font color="Green">E78</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Maroon">G</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> )
 ;<br/>

<a NAME="E18:9_3"/><b>then </b><i><font color="Green">E79</font></i>: 
( <font color="Maroon">G</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1)</span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a>  )
 
<b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E19:9_3"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>set </b><font color="Maroon">pp</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E20:9_3"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E21:9_3"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E22:9_3"/><b>then </b><i><font color="Green">E83</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E23:9_3"/><i><font color="Green">E84</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<a NAME="E24:9_3"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="jordan5b.html#T1" target="_self">Th1</a>, , <a class="txt" href="jordan5b.html#E1:9"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan5b.html#E4"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="goboard1.html#T16">GOBOARD1:16</a></i>;<br/>
<a NAME="E25:9_3"/><b>then </b><i><font color="Green">E86</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">pp</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E27:9_3"/><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E28:9_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 3
 
;<br/>
<a NAME="E29:9_3"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E30:9_3"/><i><font color="Green">E90</font></i>: 
1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">pp</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_3_1"/>
<font color="Maroon">pp</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E2:9_3_1"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>
<b>hence </b><a NAME="E3:9_3_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="partfun2.html#T17">PARTFUN2:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:9_3"/><i><font color="Green">E91</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pp</font> &amp; <font color="Maroon">pp</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan5b.html#E2:9"><i><font color="Green">E32</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E32:9_3"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">pp</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan5b.html#E4:9"><i><font color="Green">E36</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">pn</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>;<br/>
<b>set </b><font color="Maroon">pn1</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<a NAME="E33:9_3"/>
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E34:9_3"/><b>then </b><i><font color="Green">E95</font></i>: 
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a>  )
 
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a>, <a class="ref" href="jordan3.html#T37">JORDAN3:37</a></i>;<br/>
<a NAME="E35:9_3"/><b>then </b>
<font color="Maroon">NE</font> <a href="topreal1.html#R1">is_an_arc_of</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T31">TOPREAL1:31</a></i>;<br/>
<b>then consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE</font></span>)</span><b> such that </b><br/><a NAME="E37:9_3"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">F</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan5b.html#E8"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>
<a NAME="E38:9_3"/>
( <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E39:9_3"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan5b.html#E3:9"><i><font color="Green">E33</font></i></a>, <a class="ref" href="partfun2.html#T17">PARTFUN2:17</a></i>;<br/>
<a NAME="E40:9_3"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/>
<b>then consider </b><font color="Maroon">F'</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E42:9_3"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">F'</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">F'</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> &amp; <font color="Maroon">F'</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">Ex1</font> =  <a href="treal_1.html#K4">P[01]</a> 0,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">Ex2</font> =  <a href="treal_1.html#K4">P[01]</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">F1</font> = <font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">F1'</font> = <font color="Maroon">F'</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span>;<br/>
<a NAME="E43:9_3"/><i><font color="Green">E103</font></i>: 
(  <a href="treal_1.html#K4">P[01]</a> 0,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp;  <a href="treal_1.html#K4">P[01]</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  )
 
<b>by </b><i><a class="ref" href="treal_1.html#T20">TREAL_1:20</a></i>;<br/>
<a NAME="E44:9_3"/><b>then </b><i><font color="Green">E104</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E45:9_3"/><b>then </b><i><font color="Green">E105</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></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="E46:9_3"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> 
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E47:9_3"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E48:9_3"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/>
<a NAME="E49:9_3"/><b>then </b><i><font color="Green">E107</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E50:9_3"/><b>then </b><i><font color="Green">E108</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font>)
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E51:9_3"/><i><font color="Green">E109</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E52:9_3"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 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> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E54:9_3"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">F1</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i>, , <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#T71">TOPS_2:71</a></i>;<br/>
<b>then </b><i><font color="Green">E111</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F1</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE</font></span>)</span>
<b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font>)
<b>by </b><i>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<a NAME="E56:9_3"/><i><font color="Green">E112</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E57:9_3"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> 
 
<b>by </b><i><a class="ref" href="jordan5b.html#T7" target="_self">Th7</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E58:9_3"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F'</font>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E59:9_3"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F'</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/>
<a NAME="E60:9_3"/><b>then </b><i><font color="Green">E114</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan5b.html#T7" target="_self">Th7</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E61:9_3"/><i><font color="Green">E115</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E8:9"><i><font color="Green">E51</font></i></a>, , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E62:9_3"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 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">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">F1'</font> = <font color="Maroon">F'</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E64:9_3"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">F1'</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="ref" href="jordan5b.html#T7" target="_self">Th7</a>, , <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#T71">TOPS_2:71</a></i>;<br/>
<b>then </b><i><font color="Green">E117</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F1'</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<b>set </b><font color="Maroon">FF</font> = <font color="Maroon">F1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">F1'</font>;<br/>
<b>reconsider </b><font color="Maroon">T1</font> =  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>, <font color="Maroon">T2</font> =  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1 as    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="borsuk_1.html#K20">I[01]</a>  <b>by </b><i><a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="treal_1.html#T6">TREAL_1:6</a></i>;<br/>
<a NAME="E67:9_3"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="topreal3.html#T45">TOPREAL3:45</a></i>;<br/>
<i><font color="Green">E122</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1)</span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1)</span>)</span>

<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1)
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<a NAME="E69:9_3"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE1</font></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g11</font> = <font color="Maroon">F1</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE1</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E71:9_3"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F1'</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE1</font></span>)</span> )
 
<b>by </b><i>, , , , <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g22</font> = <font color="Maroon">F1'</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE1</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E73:9_3"/><i><font color="Green">E125</font></i>: 
( <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1</font> &amp; <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font> )
 
<b>by </b><i>, , , <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E74:9_3"/>
( 1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  &amp; 1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  )
 
;<br/>
<a NAME="E75:9_3"/><b>then </b><i><font color="Green">E126</font></i>: 
( 1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1</font> &amp; 1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font> )
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<i><font color="Green">E127</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">F1'</font></span>)</span> = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="treal_1.html#T2">TREAL_1:2</a></i>
;<br/>
<a NAME="E77:9_3"/><i><font color="Green">E128</font></i>: 
 <a href="borsuk_1.html#K20">I[01]</a>  is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i><a class="ref" href="heine.html#T11">HEINE:11</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E78:9_3"/><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE1</font> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T3">TOPMETR:3</a></i>;<br/>
<a NAME="E79:9_3"/><i><font color="Green">E130</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">F1'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> 
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<i><font color="Green">E131</font></i>: <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="treal_1.html#K2">(#)</a> </span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>
<br/>.= 
0,1 <a href="treal_1.html#K2">(#)</a> 
<b>by </b><i><a class="ref" href="treal_1.html#T16">TREAL_1:16</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>
;<br/>
<i><font color="Green">E132</font></i>: <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
<br/>.= 
 <a href="treal_1.html#K1">(#)</a> 0,1
<b>by </b><i><a class="ref" href="treal_1.html#T16">TREAL_1:16</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
;<br/>
<a NAME="E82:9_3"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i>, , <a class="txt" href="jordan5b.html#E2:9_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E83:9_3"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">F1'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan5b.html#T7" target="_self">Th7</a>, , <a class="txt" href="jordan5b.html#E3:9_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E84:9_3"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></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">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="treal_1.html#T3">TREAL_1:3</a></i>;<br/>
<a NAME="E85:9_3"/><b>then </b><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>;<br/>
<a NAME="E86:9_3"/><i><font color="Green">E136</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> iff <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:9_3_8"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_3_8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:9_3_8_1"/><b>then </b><i><font color="Green">E138</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font>) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:9_3_8_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D6">TOPREAL1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:9_3_8_1"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E6:9_3_8_1"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:9_3_8_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E9:9_3_8_1"/><i><font color="Green">E143</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan5b.html#E12:9_3"><i><font color="Green">E73</font></i></a></i>;<br/>
<a NAME="E10:9_3_8_1"/><i><font color="Green">E144</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E11:9_3_8_1"/><i><font color="Green">E145</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E12:9_3_8_1"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan5b.html#E14:9_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E13:9_3_8_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">f</font> is <a href="topreal1.html#V3">s.n.c.</a>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E14:9_3_8_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">f</font> is <a href="topreal1.html#V2">unfolded</a>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_8_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_3_8_1_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:9_3_8_1_1"/><i><font color="Green">E150</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan5b.html#E14:9_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:9_3_8_1_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="txt" href="jordan5b.html#E14:9_3"><i><font color="Green">E75</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:9_3_8_1_1"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>set </b><font color="Maroon">p1'</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>;<br/><b>set </b><font color="Maroon">p2'</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E5:9_3_8_1_1"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:9_3_8_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:9_3_8_1_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan5b.html#E14:9_3"><i><font color="Green">E75</font></i></a>, , <a class="txt" href="jordan5b.html#E20:9_3"><i><font color="Green">E82</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:9_3_8_1_1"/><b>then </b><i><font color="Green">E155</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:9_3_8_1_1"/><b>then </b><i><font color="Green">E156</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/><a NAME="E10:9_3_8_1_1"/><i><font color="Green">E157</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan5b.html#E24:9_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/><i><font color="Green">E158</font></i>:  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> = 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan5b.html#E14:9_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan5b.html#E22:9_3"><i><font color="Green">E83</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="jordan5b.html#E14:9_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan5b.html#E25:9_3"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan5b.html#E27:9_3"><i><font color="Green">E87</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
;<br/><a NAME="E12:9_3_8_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E13:9_3_8_1_1"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<font color="Maroon">k</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan5b.html#E18:9_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan5b.html#E23:9_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan5b.html#E28:9_3"><i><font color="Green">E88</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/><a NAME="E14:9_3_8_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E15:9_3_8_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan5b.html#E10:9_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan5b.html#E11:9_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan5b.html#E13:9_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:9_3_8_1"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan5b.html#E17:9_3"><i><font color="Green">E78</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E17:9_3_8_1"/><i><font color="Green">E160</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="txt" href="jordan5b.html#E14:9_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan5b.html#E17:9_3"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>set </b><font color="Maroon">p1'</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">p2'</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<a NAME="E18:9_3_8_1"/><i><font color="Green">E161</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E19:9_3_8_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E20:9_3_8_1"/><b>then </b><i><font color="Green">E162</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E21:9_3_8_1"/><b>then </b><i><font color="Green">E163</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/>
<a NAME="E22:9_3_8_1"/><i><font color="Green">E164</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E23:9_3"><i><font color="Green">E84</font></i></a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/>
<i><font color="Green">E165</font></i>:  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan5b.html#E20:9_3"><i><font color="Green">E82</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="txt" href="jordan5b.html#E22:9_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan5b.html#E24:9_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan5b.html#E25:9_3"><i><font color="Green">E86</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
;<br/>
<a NAME="E24:9_3_8_1"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E25:9_3_8_1"/><b>then </b><i><font color="Green">E166</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E10:9_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan5b.html#E11:9_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan5b.html#E13:9_3"><i><font color="Green">E74</font></i></a>, , <a class="txt" href="jordan5b.html#E27:9_3"><i><font color="Green">E87</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E26:9_3_8_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E27:9_3_8_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="txt" href="jordan5b.html#E19:9_3"><i><font color="Green">E80</font></i></a>, <a class="ref" href="topreal1.html#D8">TOPREAL1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E28:9_3_8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jordan5b.html#E28:9_3"><i><font color="Green">E88</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>assume </b><a NAME="E2:9_3_8"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 ;<br/>

<a NAME="E3:9_3_8"/><b>then </b><i><font color="Green">E168</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E4:9_3_8"/><i><font color="Green">E169</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E5:9_3_8"/><b>then </b>
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:9_3_8"/><b>then </b><i><font color="Green">E170</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E10:9_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/>
<a NAME="E7:9_3_8"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E8:9_3_8"/><b>then </b><i><font color="Green">E171</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E12:9_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan5b.html#E13:9_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="topreal1.html#T27">TOPREAL1:27</a></i>;<br/>
<a NAME="E9:9_3_8"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E10:9_3_8"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E11:9_3_8"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> ) </span>}</span> 
 
;<br/>
<a NAME="E12:9_3_8"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan5b.html#E14:9_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E13:9_3_8"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font>)
 
<b>by </b><i><a class="ref" href="topreal1.html#D6">TOPREAL1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E14:9_3_8"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan5b.html#E11:9_3"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E87:9_3"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F1'</font>
 
<b>by </b><i>, <a class="txt" href="jordan5b.html#E7:9_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E88:9_3"/><b>then </b><i><font color="Green">E172</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F1'</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E89:9_3"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">F1</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="txt" href="jordan5b.html#E6:9_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan5b.html#E8:9_3"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>;<br/>
<a NAME="E90:9_3"/><b>then </b><i><font color="Green">E174</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">F1'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font>) <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="jordan5b.html#E10:9_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="topmetr2.html#T3">TOPMETR2:3</a></i>;<br/>
<a NAME="E91:9_3"/><b>then </b><i><font color="Green">E175</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">F1'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1)</span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E92:9_3"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">F1'</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a>  &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">F1'</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1)</span>)</span> )
 
<b>by </b><i>, , , , , <a class="txt" href="jordan5b.html#E10:9_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan5b.html#E11:9_3"><i><font color="Green">E72</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="topmetr2.html#T3">TOPMETR2:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">FF</font> = <font color="Maroon">F1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">F1'</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE1</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E94:9_3"/><i><font color="Green">E176</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E95:9_3"/>
( 0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  &amp; 1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  )
 
;<br/>
<a NAME="E96:9_3"/><b>then </b><i><font color="Green">E177</font></i>: 
( 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> &amp; 1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> &amp;  <a href="treal_1.html#K4">P[01]</a> 0,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="treal_1.html#K4">P[01]</a> 0,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span> is <a href="pre_topc.html#V5">continuous</a> )
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<i><font color="Green">E178</font></i>: <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
<br/>.= 
 <a href="treal_1.html#K1">(#)</a> 0,1
<b>by </b><i><a class="ref" href="treal_1.html#T16">TREAL_1:16</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
;<br/>
<i><font color="Green">E179</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="jordan5b.html#E14:9_3"><i><font color="Green">E75</font></i></a>, , <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
<br/>.= 
0
<b>by </b><i>, , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
;<br/>
<i><font color="Green">E180</font></i>: <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
<br/>.= 
 <a href="treal_1.html#K1">(#)</a> 0,1
<b>by </b><i><a class="ref" href="treal_1.html#T16">TREAL_1:16</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
;<br/>
<i><font color="Green">E181</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="jordan5b.html#E14:9_3"><i><font color="Green">E75</font></i></a>, , <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> 2
<b>by </b><i><a class="txt" href="jordan5b.html#E2:9_3"><i><font color="Green">E65</font></i></a>, , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
;<br/>
<i><font color="Green">E182</font></i>: <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>
<br/>.= 
0,1 <a href="treal_1.html#K2">(#)</a> 
<b>by </b><i><a class="ref" href="treal_1.html#T16">TREAL_1:16</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>
;<br/>
<a NAME="E102:9_3"/><i><font color="Green">E183</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F1'</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#T45">FUNCT_2:45</a></i>;<br/>
<a NAME="E103:9_3"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan5b.html#E7:9_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/>
<a NAME="E104:9_3"/><i><font color="Green">E185</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 holds <br/> not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_3_14"/><i><font color="Green">E186</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/>

<b>assume </b><a NAME="E2:9_3_14"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font>
 ;<br/>

<a NAME="E3:9_3_14"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan5b.html#E24:9_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:9_3_14"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan5b.html#E8:9_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan5b.html#E24:9_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E105:9_3"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">FF</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F1'</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_15"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:9_3_15"/>
<font color="Maroon">FF</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F1'</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:9_3_15_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_3_15_1"/><i><font color="Green">E189</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FF</font> &amp; <font color="Maroon">x</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">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:9_3_15_1"/>
<font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="txt" href="jordan5b.html#E25:9_3"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/>
<b>hence </b><a NAME="E5:9_3_15_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1'</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="txt" href="jordan5b.html#E25:9_3"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


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

<b>thus </b><a NAME="E2:9_3_15"/>
<font color="Maroon">F1'</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">FF</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:9_3_15_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1'</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_3_15_2"/><i><font color="Green">E190</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font> &amp; <font color="Maroon">x</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">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:9_3_15_2"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FF</font>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E25:9_3"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_4.html#T13">FUNCT_4:13</a></i>;<br/>
<a NAME="E5:9_3_15_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E25:9_3"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/>
<b>hence </b><a NAME="E6:9_3_15_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan5b.html#E25:9_3"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan5b.html#E27:9_3"><i><font color="Green">E87</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


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


</div><b>end;</b></div>
<b>set </b><font color="Maroon">GF</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font>;<br/>
<a NAME="E106:9_3"/><i><font color="Green">E193</font></i>: 
( 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FF</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FF</font> )
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E107:9_3"/>
0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span> 
 
;<br/>
<a NAME="E108:9_3"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E109:9_3"/><b>then </b>
not 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E23:9_3"><i><font color="Green">E84</font></i></a></i>;<br/>
<b>then </b><i><font color="Green">E194</font></i>: <font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> 0 = 
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E111:9_3"/>
1 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<a NAME="E112:9_3"/><b>then </b><i><font color="Green">E195</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E196</font></i>: <font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">F1'</font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan5b.html#T7" target="_self">Th7</a>, , <a class="txt" href="jordan5b.html#E28:9_3"><i><font color="Green">E88</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/>
<a NAME="E114:9_3"/><i><font color="Green">E197</font></i>: 
( 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 0 )
 
<b>by </b><i>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<i><font color="Green">E198</font></i>: <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
<br/>.= 
0
<b>by </b><i>, <a class="txt" href="jordan5b.html#E30:9_3"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
;<br/>
<a NAME="E116:9_3"/><b>then </b><i><font color="Green">E199</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="jordan5b.html#E25:9_3"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan5b.html#E27:9_3"><i><font color="Green">E87</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E117:9_3"/><i><font color="Green">E200</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1 )
 
<b>by </b><i>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<i><font color="Green">E201</font></i>: <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
<br/>.= 
1
<b>by </b><i>, , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
;<br/>
<a NAME="E119:9_3"/><b>then </b><i><font color="Green">E202</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="jordan5b.html#E25:9_3"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan5b.html#E29:9_3"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">ppp</font> = 1 <a href="real_1.html#K6">/</a> 2 as   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E121:9_3"/>
 <a href="pcomps_1.html#K4">TopSpaceMetr</a> <a href="metric_1.html#K8">RealSpace</a>  <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T38">PCOMPS_1:38</a></i>;<br/>
<a NAME="E122:9_3"/><b>then </b><i><font color="Green">E204</font></i>: 
 <a href="borsuk_1.html#K20">I[01]</a>  <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T3">TOPMETR:3</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E123:9_3"/><i><font color="Green">E205</font></i>: 
( <font color="Maroon">T1</font> is <a href="compts_1.html#V2">compact</a> &amp; <font color="Maroon">T2</font> is <a href="compts_1.html#V2">compact</a> )
 
<b>by </b><i><a class="ref" href="heine.html#T11">HEINE:11</a></i>;<br/>
<a NAME="E124:9_3"/>
( <font color="Maroon">F1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">F1'</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE1</font> &amp; <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE1</font> )
 
<b>by </b><i>, , , , <a class="ref" href="topmetr.html#T5">TOPMETR:5</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E125:9_3"/><b>then </b><i><font color="Green">E206</font></i>: 
( <font color="Maroon">g11</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g22</font> is <a href="pre_topc.html#V5">continuous</a> )
 
<b>by </b><i><a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/>
<a NAME="E126:9_3"/><i><font color="Green">E207</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></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="E127:9_3"/><i><font color="Green">E208</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font> <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">1 <a href="real_1.html#K6">/</a> 2</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="E128:9_3"/><b>then </b><i><font color="Green">E209</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> 
 
<b>by </b><i>, <a class="ref" href="treal_1.html#T2">TREAL_1:2</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<a NAME="E129:9_3"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">ppp</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="topmetr2.html#T1">TOPMETR2:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">FF</font> = <font color="Maroon">FF</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE1</font></span>)</span> <b>by </b><i><a class="txt" href="jordan5b.html#E6:9_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan5b.html#E7:9_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan5b.html#E37:9_3"><i><font color="Green">E96</font></i></a>, , <a class="txt" href="jordan5b.html#E42:9_3"><i><font color="Green">E98</font></i></a>, , <a class="ref" href="topmetr2.html#T4">TOPMETR2:4</a></i>;<br/>
<a NAME="E131:9_3"/><i><font color="Green">E210</font></i>: 
( <font color="Maroon">F1</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">F1'</font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i>, , <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E132:9_3"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font></span>)</span> holds <br/><font color="Maroon">F1'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_20"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_3_20"/><i><font color="Green">E213</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font></span>)</span> )
 ;<br/>

<b>assume </b><a NAME="E2:9_3_20"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">F1'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/>

<a NAME="E3:9_3_20"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> 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">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E43:9_3"><i><font color="Green">E103</font></i></a></i>;<br/>
<a NAME="E4:9_3_20"/><b>then </b><i><font color="Green">E215</font></i>: 
<font color="Maroon">F1'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<a NAME="E5:9_3_20"/><i><font color="Green">E216</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1</font> &amp; not <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font> )
 
<b>by </b><i><a class="txt" href="jordan5b.html#E43:9_3"><i><font color="Green">E103</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:9_3_20"/><b>then </b>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> 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> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 
;<br/>
<a NAME="E7:9_3_20"/><b>then </b>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">NE</font>
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<a NAME="E8:9_3_20"/><b>then </b>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">NE</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E44:9_3"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan5b.html#E45:9_3"><i><font color="Green">E105</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:9_3_20"/><b>then </b>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan5b.html#E9:9_3"><i><font color="Green">E70</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:9_3_20"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan5b.html#E6:9_3"><i><font color="Green">E67</font></i></a>, , <a class="txt" href="jordan5b.html#E47:9_3"><i><font color="Green">E106</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E133:9_3"/><b>then </b>
<font color="Maroon">FF</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="topmetr2.html#T2">TOPMETR2:2</a></i>;<br/>
<a NAME="E134:9_3"/><b>then </b>
( <font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">FF</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  )
 
<b>by </b><i>, <a class="txt" href="jordan5b.html#E9:9"><i><font color="Green">E62</font></i></a>, , <a class="txt" href="jordan5b.html#E1:9_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan5b.html#E13:9_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="compts_1.html#T26">COMPTS_1:26</a>, <a class="ref" href="tops_2.html#T70">TOPS_2:70</a></i>;<br/>
<a NAME="E135:9_3"/><b>then </b><i><font color="Green">E217</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="ref" href="tops_2.html#T71">TOPS_2:71</a></i>;<br/>
<a NAME="E136:9_3"/><b>then </b><i><font color="Green">E218</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E137:9_3"/><i><font color="Green">E219</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a>  &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="jordan5b.html#E43:9_3"><i><font color="Green">E103</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E138:9_3"/><b>then </b><i><font color="Green">E220</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a>  &amp;  <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">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a>  )
 
<b>by </b><i><a class="ref" href="tops_2.html#T62">TOPS_2:62</a></i>;<br/>
<i><font color="Green">E221</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1)</span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">FF</font>
<b>by </b><i>, , <a class="txt" href="jordan5b.html#E12:9_3"><i><font color="Green">E73</font></i></a></i>
;<br/>
<a NAME="E140:9_3"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">G</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">FF</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_22"><b>proof </b></a><div class="add">

<font color="Maroon">G</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> = 
<font color="Maroon">FF</font> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font>
<b>by </b><i>, <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>
<br/>.= 
<font color="Maroon">FF</font>
<b>by </b><i><a class="txt" href="jordan5b.html#E49:9_3"><i><font color="Green">E107</font></i></a>, <a class="ref" href="relat_1.html#T80">RELAT_1:80</a></i>
;<br/>
<b>hence </b><a NAME="E2:9_3_22"/>
<font color="Maroon">G</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">FF</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E141:9_3"/>
1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E142:9_3"/><b>then </b><i><font color="Green">E223</font></i>: 
1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E45:9_3"><i><font color="Green">E105</font></i></a></i>;<br/>
<a NAME="E143:9_3"/><b>then </b><i><font color="Green">E224</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <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"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<i><font color="Green">E225</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan5b.html#E51:9_3"><i><font color="Green">E109</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">pp</font>
<b>by </b><i>, <a class="txt" href="jordan5b.html#E7:9_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
;<br/>
<a NAME="E145:9_3"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">pp</font>,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_24"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:9_3_24"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">pp</font>,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:9_3_24_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">pp</font>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<a NAME="E2:9_3_24_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">pp</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">l1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:9_3_24_1"/><i><font color="Green">E226</font></i>: 
( <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">pp</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l1</font> &amp; <font color="Maroon">l1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<a NAME="E5:9_3_24_1"/><i><font color="Green">E227</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pp</font>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E54:9_3"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan5b.html#E55:9_3"><i><font color="Green">E111</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<b>set </b><font color="Maroon">cos</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l1</font>;<br/>
<a NAME="E6:9_3_24_1"/>
<font color="Maroon">l1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E47:9_3"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jordan5b.html#E56:9_3"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jordan5b.html#E58:9_3"><i><font color="Green">E113</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E7:9_3_24_1"/><b>then </b><i><font color="Green">E229</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l1</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">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:9_3_24_1"/><b>then </b><i><font color="Green">E230</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<a NAME="E9:9_3_24_1"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">l1</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"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E45:9_3"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan5b.html#E56:9_3"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jordan5b.html#E58:9_3"><i><font color="Green">E113</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<i><font color="Green">E232</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l1</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan5b.html#E45:9_3"><i><font color="Green">E105</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
<br/>.= 
<font color="Maroon">l1</font>
<b>by </b><i><a class="txt" href="jordan5b.html#E45:9_3"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan5b.html#E64:9_3"><i><font color="Green">E116</font></i></a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">cos</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jordan5b.html#E61:9_3"><i><font color="Green">E115</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">A3</font> = <font color="Maroon">cos</font>, <font color="Maroon">A4</font> = 1, <font color="Maroon">A5</font> = 1 <a href="real_1.html#K6">/</a> 2 as   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a>  <b>by </b><i><a class="txt" href="jordan5b.html#E60:9_3"><i><font color="Green">E114</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A3</font>, <font color="Maroon">A2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A4</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a>  ;<br/>
<b>reconsider </b><font color="Maroon">Fhc</font> = <font color="Maroon">A1</font>, <font color="Maroon">Fh0</font> = <font color="Maroon">A2</font>, <font color="Maroon">Fh12</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A5</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E15:9_3_24_1"/><i><font color="Green">E236</font></i>: 
<font color="Maroon">cos</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_24_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_3_24_1_2"/>
<font color="Maroon">cos</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/>

<a NAME="E2:9_3_24_1_2"/><b>then </b>
<font color="Maroon">Fhc</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">Fh0</font>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E32:9_3"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jordan5b.html#E34:9_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan5b.html#E44:9_3"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan5b.html#E45:9_3"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jordan5a.html#T17">JORDAN5A:17</a></i>;<br/>
<b>hence </b><a NAME="E3:9_3_24_1_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan5b.html#E29:9_3"><i><font color="Green">E89</font></i></a>, , <a class="txt" href="jordan5b.html#E56:9_3"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jordan5b.html#E65:9_3"><i><font color="Green">E117</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:9_3_24_1"/>
<font color="Maroon">cos</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_24_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_3_24_1_3"/>
<font color="Maroon">cos</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/>

<a NAME="E2:9_3_24_1_3"/><b>then </b>
<font color="Maroon">Fhc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">Fh12</font>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E32:9_3"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jordan5b.html#E34:9_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan5b.html#E44:9_3"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan5b.html#E45:9_3"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jordan5a.html#T17">JORDAN5A:17</a></i>;<br/>
<b>hence </b><a NAME="E3:9_3_24_1_3"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan5b.html#E22:9_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan5b.html#E56:9_3"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jordan5b.html#E65:9_3"><i><font color="Green">E117</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:9_3_24_1"/><b>then </b>
<font color="Maroon">cos</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E18:9_3_24_1"/><b>then </b>
<font color="Maroon">cos</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">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E19:9_3_24_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan5b.html#E45:9_3"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan5b.html#E56:9_3"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jordan5b.html#E60:9_3"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan5b.html#E65:9_3"><i><font color="Green">E117</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


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

<b>thus </b><a NAME="E2:9_3_24"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">pp</font>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_24_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:9_3_24_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_3_24_2"/><i><font color="Green">E237</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> &amp; <font color="Maroon">x</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">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:9_3_24_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan5b.html#E56:9_3"><i><font color="Green">E112</font></i></a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">l1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:9_3_24_2"/><i><font color="Green">E238</font></i>: 
( <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l1</font> &amp; <font color="Maroon">l1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<b>reconsider </b><font color="Maroon">ll</font> = <font color="Maroon">l1</font>, <font color="Maroon">pol</font> = 1 <a href="real_1.html#K6">/</a> 2 as   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a>  <b>by </b><i><a class="txt" href="jordan5b.html#E58:9_3"><i><font color="Green">E113</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_2.html#K3">.</a> <a href="borsuk_1.html#K22">1[01]</a> , <font color="Maroon">A2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">ll</font>, <font color="Maroon">A3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">pol</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a>  ;<br/>
<b>reconsider </b><font color="Maroon">A4</font> = <font color="Maroon">A1</font>, <font color="Maroon">A5</font> = <font color="Maroon">A2</font>, <font color="Maroon">A6</font> = <font color="Maroon">A3</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E10:9_3_24_2"/><i><font color="Green">E241</font></i>: 
<font color="Maroon">A4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">A5</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_24_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_24_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_3_24_2_1_1"/>
( 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">l1</font> or 1 <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_24_2_1_1_1"><b>suppose </b></a><a NAME="E1:9_3_24_2_1_1_1"/>
1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">l1</font>
 ;<br/><div class="add"><a NAME="E2:9_3_24_2_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">l1</font>
 <b>by </b><i><a class="txt" href="jordan5b.html#E58:9_3"><i><font color="Green">E113</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:9_3_24_2_1_1_1"/>
<font color="Maroon">A4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">A5</font>
 <b>by </b><i><a class="txt" href="jordan5b.html#E32:9_3"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jordan5b.html#E34:9_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan5b.html#E44:9_3"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan5b.html#E45:9_3"><i><font color="Green">E105</font></i></a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="jordan5a.html#T17">JORDAN5A:17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_24_2_1_1_2"><b>suppose </b></a><a NAME="E1:9_3_24_2_1_1_2"/>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_3_24_2_1_1_2"/>
<font color="Maroon">A4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">A5</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E11:9_3_24_2"/>
<font color="Maroon">A5</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">A6</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_24_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_24_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_3_24_2_2_1"/>
( <font color="Maroon">l1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 or <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_24_2_2_1_1"><b>suppose </b></a><a NAME="E1:9_3_24_2_2_1_1"/>
<font color="Maroon">l1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:9_3_24_2_2_1_1"/><b>then </b>
<font color="Maroon">l1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="jordan5b.html#E58:9_3"><i><font color="Green">E113</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:9_3_24_2_2_1_1"/>
<font color="Maroon">A5</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">A6</font>
 <b>by </b><i><a class="txt" href="jordan5b.html#E32:9_3"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jordan5b.html#E34:9_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan5b.html#E44:9_3"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan5b.html#E45:9_3"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jordan5a.html#T17">JORDAN5A:17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_24_2_2_1_2"><b>suppose </b></a><a NAME="E1:9_3_24_2_2_1_2"/>
<font color="Maroon">l1</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_3_24_2_2_1_2"/>
<font color="Maroon">A5</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">A6</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E12:9_3_24_2"/><b>then </b>
<font color="Maroon">A5</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">pp</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan5b.html#E34:9_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan5b.html#E55:9_3"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan5b.html#E60:9_3"><i><font color="Green">E114</font></i></a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a></i>;<br/>
<b>hence </b><a NAME="E13:9_3_24_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">pp</font>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan5b.html#E56:9_3"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jordan5b.html#E58:9_3"><i><font color="Green">E113</font></i></a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E146:9_3"/><b>then </b><i><font color="Green">E242</font></i>: 
<font color="Maroon">G</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">pp</font>,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E20:9_3"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan5b.html#E24:9_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan5b.html#E50:9_3"><i><font color="Green">E108</font></i></a>, <a class="ref" href="relat_1.html#T159">RELAT_1:159</a></i>;<br/>
<a NAME="E147:9_3"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">p1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p1</font> &amp; <font color="Olive">p1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> &amp; <font color="Olive">p2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">p1</font>,<font color="Olive">p2</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_25"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_25_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_3_25_1"/>
( <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 or <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_25_1_1"><b>suppose </b></a><a NAME="E1:9_3_25_1_1"/>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:9_3_25_1_1"/><i><font color="Green">E245</font></i>: 
( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, , </i>;<br/><b>set </b><font color="Maroon">f1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font>;<br/><b>set </b><font color="Maroon">f2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font>;<br/><a NAME="E4:9_3_25_1_1"/><i><font color="Green">E248</font></i>: 
(  <a href="treal_1.html#K4">P[01]</a> 0,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span> is <a href="pre_topc.html#V5">continuous</a> &amp;  <a href="treal_1.html#K4">P[01]</a> 0,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span> )
 <b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E5:9_3_25_1_1"/><b>then </b><i><font color="Green">E249</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> )
 <b>by </b><i><a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/> <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> 
<b>by </b><i>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> 

;<br/><a NAME="E7:9_3_25_1_1"/><b>then </b><i><font color="Green">E250</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan5b.html#E58:9_3"><i><font color="Green">E113</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><a NAME="E8:9_3_25_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan5b.html#E60:9_3"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan5b.html#E61:9_3"><i><font color="Green">E115</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/><a NAME="E9:9_3_25_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> 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> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> 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> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> )
 ;<br/><a NAME="E10:9_3_25_1_1"/><b>then </b><i><font color="Green">E251</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></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/><b>then reconsider </b><font color="Maroon">f1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font>, <font color="Maroon">f2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><b>reconsider </b><font color="Maroon">r1'</font> = <font color="Maroon">r1</font>, <font color="Maroon">r2'</font> = <font color="Maroon">r2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span> <b>by </b><i><a class="txt" href="jordan5b.html#E58:9_3"><i><font color="Green">E113</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/><a NAME="E13:9_3_25_1_1"/>
<span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, <a class="ref" href="tops_2.html#T70">TOPS_2:70</a></i>;<br/><a NAME="E14:9_3_25_1_1"/><b>then </b><i><font color="Green">E254</font></i>: 
( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1'</font> &amp; <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">r2'</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a> &amp; <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i><a class="txt" href="jordan5b.html#E17:9_3"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan5b.html#E19:9_3"><i><font color="Green">E80</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E15:9_3_25_1_1"/><b>then </b><i><font color="Green">E255</font></i>: 
<font color="Maroon">f1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="txt" href="jordan5b.html#E58:9_3"><i><font color="Green">E113</font></i></a>, <a class="ref" href="jordan5a.html#T16">JORDAN5A:16</a></i>;<br/><a NAME="E16:9_3_25_1_1"/><i><font color="Green">E256</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="treal_1.html#T1">TREAL_1:1</a></i>;<br/><a NAME="E17:9_3_25_1_1"/><i><font color="Green">E257</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f2</font> )
 <b>by </b><i><a class="txt" href="jordan5b.html#E60:9_3"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan5b.html#E61:9_3"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan5b.html#E64:9_3"><i><font color="Green">E116</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/><a NAME="E18:9_3_25_1_1"/><i><font color="Green">E258</font></i>: 
( <font color="Maroon">f1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  &amp; <font color="Maroon">f2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan5b.html#E65:9_3"><i><font color="Green">E117</font></i></a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">ff1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E20:9_3_25_1_1"/><i><font color="Green">E260</font></i>: 
( <font color="Maroon">ff1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ff1</font> &amp; <font color="Maroon">ff1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 ;<br/><b>consider </b><font color="Maroon">ff2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E22:9_3_25_1_1"/><i><font color="Green">E262</font></i>: 
( <font color="Maroon">ff2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ff2</font> &amp; <font color="Maroon">ff2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 <b>by </b><i><a class="txt" href="jordan5b.html#E68:9_3"><i><font color="Green">E122</font></i></a></i>;<br/><a NAME="E23:9_3_25_1_1"/><i><font color="Green">E263</font></i>: 
<span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="txt" href="jordan5b.html#E2:9_3"><i><font color="Green">E65</font></i></a>, , , <a class="txt" href="jordan5b.html#E67:9_3"><i><font color="Green">E121</font></i></a>, , , <a class="ref" href="jordan5a.html#T21">JORDAN5A:21</a></i>;<br/><a NAME="E24:9_3_25_1_1"/>
<font color="Maroon">F1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_25_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:9_3_25_1_1_2"/>
<font color="Maroon">F1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">FF</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="rcomp_1.html#K1">.]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_25_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:9_3_25_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_3_25_1_1_2_1"/><i><font color="Green">E264</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_25_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_25_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_3_25_1_1_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_25_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:9_3_25_1_1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:9_3_25_1_1_2_1_1_1_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font>
 <b>by </b><i>, <a class="txt" href="jordan5b.html#E23:9_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan5b.html#E75:9_3"><i><font color="Green">E126</font></i></a></i>;<br/><b>hence </b><a NAME="E3:9_3_25_1_1_2_1_1_1_1"/>
<font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_25_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:9_3_25_1_1_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_3_25_1_1_2_1_1_1_2"/>
<font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="jordan5b.html#E6:9_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan5b.html#E7:9_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E5:9_3_25_1_1_2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FF</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="txt" href="jordan5b.html#E75:9_3"><i><font color="Green">E126</font></i></a>, <a class="ref" href="funct_4.html#T13">FUNCT_4:13</a></i>;<br/>
<b>hence </b><a NAME="E6:9_3_25_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


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

<b>thus </b><a NAME="E2:9_3_25_1_1_2"/>
<font color="Maroon">FF</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_25_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:9_3_25_1_1_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_3_25_1_1_2_2"/><i><font color="Green">E265</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FF</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:9_3_25_1_1_2_2"/><i><font color="Green">E266</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E65:9_3"><i><font color="Green">E117</font></i></a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_25_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_25_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_3_25_1_1_2_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_25_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:9_3_25_1_1_2_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:9_3_25_1_1_2_2_1_1_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font>
 <b>by </b><i><a class="txt" href="jordan5b.html#E23:9_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan5b.html#E75:9_3"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jordan5b.html#E76:9_3"><i><font color="Green">E127</font></i></a></i>;<br/><b>hence </b><a NAME="E3:9_3_25_1_1_2_2_1_1_1"/>
<font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_25_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:9_3_25_1_1_2_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_3_25_1_1_2_2_1_1_2"/>
<font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="jordan5b.html#E6:9_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan5b.html#E7:9_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:9_3_25_1_1_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="txt" href="jordan5b.html#E75:9_3"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jordan5b.html#E76:9_3"><i><font color="Green">E127</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


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


</div><b>end;</b></div><a NAME="E25:9_3_25_1_1"/><b>then </b><i><font color="Green">E267</font></i>: 
<font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan5b.html#E73:9_3"><i><font color="Green">E125</font></i></a>, <a class="ref" href="relat_1.html#T159">RELAT_1:159</a></i>;<br/><b>set </b><font color="Maroon">e1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f1</font>;<br/><b>set </b><font color="Maroon">e2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f2</font>;<br/><a NAME="E26:9_3_25_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a>  )
 <b>by </b><i><a class="txt" href="jordan5b.html#E65:9_3"><i><font color="Green">E117</font></i></a>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E27:9_3_25_1_1"/><b>then </b><i><font color="Green">E270</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">l1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E29:9_3_25_1_1"/><i><font color="Green">E271</font></i>: 
( <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l1</font> &amp; <font color="Maroon">l1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>consider </b><font color="Maroon">l2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E31:9_3_25_1_1"/><i><font color="Green">E273</font></i>: 
( <font color="Maroon">l2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l2</font> &amp; <font color="Maroon">l2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan5b.html#E76:9_3"><i><font color="Green">E127</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">f1'</font> = <font color="Maroon">f1</font>, <font color="Maroon">f2'</font> = <font color="Maroon">f2</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a>  <b>by </b><i><a class="txt" href="jordan5b.html#E65:9_3"><i><font color="Green">E117</font></i></a>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/><a NAME="E33:9_3_25_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f1'</font> &amp; <font color="Maroon">l2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2'</font> )
 <b>by </b><i><a class="txt" href="jordan5b.html#E25:9_3"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan5b.html#E27:9_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan5b.html#E29:9_3"><i><font color="Green">E89</font></i></a>, <a class="txt" href="jordan5b.html#E31:9_3"><i><font color="Green">E91</font></i></a>, , <a class="txt" href="jordan5b.html#E77:9_3"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan5b.html#E78:9_3"><i><font color="Green">E129</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E34:9_3_25_1_1"/><b>then </b><i><font color="Green">E276</font></i>: 
( <font color="Maroon">l1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">l2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l1</font> &amp; <font color="Maroon">l1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l2</font> &amp; <font color="Maroon">l2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan5b.html#E44:9_3"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan5b.html#E45:9_3"><i><font color="Green">E105</font></i></a>, , <a class="txt" href="jordan5b.html#E77:9_3"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan5b.html#E78:9_3"><i><font color="Green">E129</font></i></a>, <a class="ref" href="jordan5a.html#T17">JORDAN5A:17</a></i>;<br/><a NAME="E35:9_3_25_1_1"/><i><font color="Green">E277</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">f1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">l2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f2</font> )
 <b>by </b><i><a class="txt" href="jordan5b.html#E43:9_3"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan5b.html#E58:9_3"><i><font color="Green">E113</font></i></a>, , <a class="txt" href="jordan5b.html#E77:9_3"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan5b.html#E78:9_3"><i><font color="Green">E129</font></i></a>, <a class="ref" href="jordan5a.html#T16">JORDAN5A:16</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E36:9_3_25_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan5b.html#E65:9_3"><i><font color="Green">E117</font></i></a>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><b>then </b><i><font color="Green">E278</font></i>: <font color="Maroon">G</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">l1</font>,<font color="Maroon">l2</font></span><a href="rcomp_1.html#K1">.]</a></span> = 
<font color="Maroon">G</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">FF</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="jordan5b.html#E32:9_3"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jordan5b.html#E34:9_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan5b.html#E43:9_3"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan5b.html#E80:9_3"><i><font color="Green">E131</font></i></a>, <a class="ref" href="jordan5a.html#T21">JORDAN5A:21</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>
<br/>.= 
<font color="Maroon">FF</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="jordan5b.html#E50:9_3"><i><font color="Green">E108</font></i></a>, <a class="ref" href="relat_1.html#T159">RELAT_1:159</a></i>
;<br/><a NAME="E38:9_3_25_1_1"/><i><font color="Green">E279</font></i>: 
<font color="Maroon">FF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_25_1_1_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_25_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_3_25_1_1_4_1"/>
( <font color="Maroon">f1'</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2 or <font color="Maroon">f1'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_25_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:9_3_25_1_1_4_1_1"/><i><font color="Green">E280</font></i>: 
<font color="Maroon">f1'</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><b>then </b><font color="Maroon">FF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f1'</font> = 
<font color="Maroon">F1'</font> <a href="funct_1.html#K1">.</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="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">F'</font> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i>, <a class="txt" href="jordan5b.html#E18:9_3"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1'</font>
<b>by </b><i>, <a class="ref" href="jordan5b.html#T7" target="_self">Th7</a>, <a class="txt" href="jordan5b.html#E2:9_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan5b.html#E60:9_3"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan5b.html#E61:9_3"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan5b.html#E83:9_3"><i><font color="Green">E134</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
;<br/><b>hence </b><a NAME="E3:9_3_25_1_1_4_1_1"/>
<font color="Maroon">FF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1'</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_25_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:9_3_25_1_1_4_1_2"/>
<font color="Maroon">f1'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:9_3_25_1_1_4_1_2"/><b>then </b>
not <font color="Maroon">f1'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font>
 <b>by </b><i><a class="txt" href="jordan5b.html#E23:9_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan5b.html#E65:9_3"><i><font color="Green">E117</font></i></a></i>;<br/><b>then </b><font color="Maroon">FF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f1'</font> = 
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f1'</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f1'</font></span>)</span>
<b>by </b><i>, <a class="txt" href="jordan5b.html#E65:9_3"><i><font color="Green">E117</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1'</font>
<b>by </b><i><a class="txt" href="jordan5b.html#E60:9_3"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan5b.html#E61:9_3"><i><font color="Green">E115</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
;<br/><b>hence </b><a NAME="E4:9_3_25_1_1_4_1_2"/>
<font color="Maroon">FF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1'</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E39:9_3_25_1_1"/><i><font color="Green">E281</font></i>: 
<font color="Maroon">FF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_25_1_1_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_25_1_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_3_25_1_1_5_1"/>
( <font color="Maroon">f2'</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2 or <font color="Maroon">f2'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 <a href="real_1.html#K6">/</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_25_1_1_5_1_1"><b>suppose </b></a><a NAME="E1:9_3_25_1_1_5_1_1"/><i><font color="Green">E282</font></i>: 
<font color="Maroon">f2'</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><b>then </b><font color="Maroon">FF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2'</font> = 
<font color="Maroon">F1'</font> <a href="funct_1.html#K1">.</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="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">F'</font> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i>, <a class="txt" href="jordan5b.html#E18:9_3"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2'</font>
<b>by </b><i>, <a class="ref" href="jordan5b.html#T7" target="_self">Th7</a>, <a class="txt" href="jordan5b.html#E2:9_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan5b.html#E60:9_3"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan5b.html#E61:9_3"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan5b.html#E85:9_3"><i><font color="Green">E135</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
;<br/><b>hence </b><a NAME="E3:9_3_25_1_1_5_1_1"/>
<font color="Maroon">FF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2'</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_25_1_1_5_1_2"><b>suppose </b></a><a NAME="E1:9_3_25_1_1_5_1_2"/>
<font color="Maroon">f2'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:9_3_25_1_1_5_1_2"/><b>then </b>
not <font color="Maroon">f2'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1'</font>
 <b>by </b><i><a class="txt" href="jordan5b.html#E23:9_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan5b.html#E65:9_3"><i><font color="Green">E117</font></i></a></i>;<br/><b>then </b><font color="Maroon">FF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2'</font> = 
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f2'</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4">P[01]</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f2'</font></span>)</span>
<b>by </b><i>, <a class="txt" href="jordan5b.html#E65:9_3"><i><font color="Green">E117</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2'</font>
<b>by </b><i><a class="txt" href="jordan5b.html#E60:9_3"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan5b.html#E61:9_3"><i><font color="Green">E115</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
;<br/><b>hence </b><a NAME="E4:9_3_25_1_1_5_1_2"/>
<font color="Maroon">FF</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2'</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E40:9_3_25_1_1"/><i><font color="Green">E283</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="txt" href="jordan5b.html#E50:9_3"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan5b.html#E58:9_3"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan5b.html#E77:9_3"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan5b.html#E82:9_3"><i><font color="Green">E133</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E41:9_3_25_1_1"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan5b.html#E50:9_3"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan5b.html#E58:9_3"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan5b.html#E78:9_3"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jordan5b.html#E83:9_3"><i><font color="Green">E134</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><b>hence </b><a NAME="E42:9_3_25_1_1"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">p1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p1</font> &amp; <font color="Olive">p1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> &amp; <font color="Olive">p2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">p1</font>,<font color="Olive">p2</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan5b.html#E58:9_3"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan5b.html#E75:9_3"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jordan5b.html#E79:9_3"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan5b.html#E81:9_3"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jordan5b.html#E85:9_3"><i><font color="Green">E135</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_3_25_1_2"><b>suppose </b></a><a NAME="E1:9_3_25_1_2"/>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2
 ;<br/><div class="add"><a NAME="E2:9_3_25_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 3
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/><a NAME="E3:9_3_25_1_2"/><b>then </b><i><font color="Green">E284</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E4:9_3_25_1_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:9_3_25_1_2"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">p1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p1</font> &amp; <font color="Olive">p1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> &amp; <font color="Olive">p2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">p1</font>,<font color="Olive">p2</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan5b.html#E2:9"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan5b.html#E5:9"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan5b.html#E7:9"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan5b.html#E56:9_3"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jordan5b.html#E58:9_3"><i><font color="Green">E113</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E148:9_3"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">p1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p1</font> &amp; <font color="Olive">p1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> &amp; <font color="Olive">p2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">p1</font>,<font color="Olive">p2</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:9"/><i><font color="Green">E285</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E11:9"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Maroon">h1</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>then consider </b><font color="Maroon">NE</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E13:9"/><i><font color="Green">E286</font></i>: 
( <font color="Maroon">NE</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">h1</font>) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">F</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">NE</font></span>)</span>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Olive">F</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> holds <br/> ex <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">p1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p1</font> &amp; <font color="Olive">p1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> &amp; <font color="Olive">p2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">p1</font>,<font color="Olive">p2</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font> &amp; <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) ) )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E14:9"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">p1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p1</font> &amp; <font color="Olive">p1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p2</font> &amp; <font color="Olive">p2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ff</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">p1</font>,<font color="Olive">p2</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">Ff</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">Ff</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, , </i>;<br/>


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