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

<b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">Ft</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:40"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">F</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E2:40"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">Ft</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E3:40"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Ft</font>
 ;<br/>

<a NAME="E4:40"/><i><font color="Green">E78</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E5:40"/><i><font color="Green">E79</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E6:40"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">Ft</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">UL</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E8:40"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">UL</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="toprealb.html#K8">Tunit_circle</a> 2
 <b>and </b><br/><a NAME="E9:40"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">UL</font> is <a href="tops_2.html#V1">open</a>
 <b>and </b><br/><a NAME="E10:40"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">U</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>  st <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UL</font> holds <br/> ex <font color="Olive">D</font> being <a href="taxonom2.html#V4">mutually-disjoint</a> <a href="tops_2.html#V1">open</a> <a href="struct_0.html#NM3">Subset-Family</a> of <a href="topalg_2.html#K3">R^1</a>  st <br/>(  <a href="setfam_1.html#K5">union</a> <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Olive">U</font> &amp; ( for <font color="Olive">d</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> holds <br/>for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Olive">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Olive">U</font></span>)</span>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Olive">d</font> holds <br/><font color="Olive">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  ) )
 <b>by </b><i/>;<br/>
<i><font color="Green">E85</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>
;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font> ex <font color="Olive">t</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  ex <font color="Olive">N</font> being non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ex <font color="Olive">Fn</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Olive">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">t</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">Fn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> &amp; <font color="Olive">Fn</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">Fn</font> &amp; <font color="Olive">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; ( for <font color="Olive">H</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Olive">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">H</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">H</font> &amp; <font color="Olive">H</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Olive">Fn</font> <a href="funct_2.html#R2">=</a> <font color="Olive">H</font> ) );<br/>
<a NAME="E12:40"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>  ex <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topalg_2.html#K3">R^1</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>, <font color="Maroon">t</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E2:40_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">t</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/>
<b>consider </b><font color="Maroon">TT</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:40_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E5:40_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">TT</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E6:40_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">TT</font> is <a href="seqm_3.html#V1">increasing</a>
 <b>and </b><br/><a NAME="E7:40_2"/><i><font color="Green">E91</font></i>: 
 ex <font color="Olive">N</font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> &amp; ( for <font color="Olive">i</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font> &amp; <font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font> holds <br/> ex <font color="Olive">Ui</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="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Olive">Ui</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UL</font> &amp; <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">N</font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">Ui</font> ) ) )
 <b>by </b><i>, , , </i>;<br/>
<b>consider </b><font color="Maroon">N</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E9:40_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E10:40_2"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">i</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font> &amp; <font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font> holds <br/> ex <font color="Olive">Ui</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="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Olive">Ui</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UL</font> &amp; <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">Ui</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">N</font> = <font color="Maroon">N</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E12:40_2"/><i><font color="Green">E94</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E13:40_2"/><i><font color="Green">E95</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">TT</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<div><i><font color="Green">E96</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1"><b>now </b></a><div class="add"><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="E1:40_2_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font>
 ;<br/><a NAME="E2:40_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E5"><i><font color="Green">Lemma59</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:40_2_1"/><b>then </b>
( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> or 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:40_2_1"/><i><font color="Green">E98</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="txt" href="topalg_5.html#E5"><i><font color="Green">Lemma59</font></i></a>, <a class="txt" href="topalg_5.html#E2"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="seqm_3.html#D1">SEQM_3:def 1</a></i>;<br/><b>assume </b><a NAME="E5:40_2_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">i</font> <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">TT</font>
 ;<br/><a NAME="E6:40_2_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E7:40_2_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</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>, <a class="txt" href="topalg_5.html#E5"><i><font color="Green">Lemma59</font></i></a>, , <a class="ref" href="seqm_3.html#D1">SEQM_3:def 1</a></i>;<br/><a NAME="E8:40_2_1"/>
<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">TT</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:40_2_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">TT</font> or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">TT</font> )
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E10:40_2_1"/>
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, , , <a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="seqm_3.html#D1">SEQM_3:def 1</a></i>;<br/><b>hence </b><a NAME="E11:40_2_1"/>
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i><a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>thus </b><a NAME="E12:40_2_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</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>, <a class="txt" href="topalg_5.html#E7"><i><font color="Green">Lemma62</font></i></a>, , <a class="txt" href="topalg_5.html#E5"><i><font color="Green">Lemma59</font></i></a>, , <a class="ref" href="seqm_3.html#D1">SEQM_3:def 1</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E102</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_2"><b>now </b></a><div class="add"><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="E1:40_2_2"/><i><font color="Green">E103</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>thus </b><a NAME="E2:40_2_2"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_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="E1:40_2_2_1"/><i><font color="Green">E104</font></i>: 
<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"><span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:40_2_2_1"/>
( <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</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>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E4:40_2_2_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:40_2_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font> implies  ex <font color="Olive">N2</font> being non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ex <font color="Olive">S</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  ex <font color="Olive">Fn</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Olive">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Olive">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  st <br/>( <font color="Olive">S</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"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">N2</font> &amp; <font color="Olive">N2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font> &amp; <font color="Olive">Fn</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">N2</font>,<font color="Olive">S</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">Fn</font> &amp; <font color="Olive">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> ) );<br/>
<a NAME="E16:40_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_3.html#T26">FINSEQ_3:26</a></i>;<br/>
<a NAME="E17:40_2"/><i><font color="Green">E106</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3"><b>proof </b></a><div class="add">

<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><b>that </b><br/><a NAME="E1:40_2_3"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>]
 <b>and </b><br/><a NAME="E2:40_2_3"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">i</font> <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">TT</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_3_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font> )
 </span><b>by </b><i><a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, <a class="ref" href="topreala.html#T23">TOPREALA:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_1"><b>suppose </b></a><a NAME="E1:40_2_3_1_1"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">S</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>;<br/><a NAME="E2:40_2_3_1_1"/><i><font color="Green">E189</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="txt" href="topalg_5.html#E9"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>;<br/><b>reconsider </b><font color="Maroon">S</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i>, , <a class="txt" href="topalg_5.html#E9"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>;<br/><b>take </b><font color="Maroon">N2</font> = <font color="Maroon">N</font>;<br/><a NAME="E4:40_2_3_1_1"/><i><font color="Green">E190</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">N2</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E5:40_2_3_1_1"/><i><font color="Green">E191</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E6:40_2_3_1_1"/><i><font color="Green">E192</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/><b>set </b><font color="Maroon">Fn</font> = <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><a NAME="E7:40_2_3_1_1"/><i><font color="Green">E193</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Ft</font>
 <b>by </b><i>, <a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E8:40_2_3_1_1"/><b>then </b><i><font color="Green">E194</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E9:40_2_3_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3">R^1</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">Fn</font> = <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  <b>by </b><i>, , , , <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><b>take </b>
<font color="Maroon">S</font>
;<br/><b>take </b>
<font color="Maroon">Fn</font>
;<br/><b>thus </b><a NAME="E11:40_2_3_1_1"/>
<font color="Maroon">S</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"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><b>thus </b><a NAME="E12:40_2_3_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E13:40_2_3_1_1"/>
<font color="Maroon">N2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 ;<br/><b>reconsider </b><font color="Maroon">K0</font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i>, , <a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E15:40_2_3_1_1"/>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span>
 <b>by </b><i>, , <a class="txt" href="topalg_5.html#E9"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>;<br/><b>then reconsider </b><font color="Maroon">S1</font> = <font color="Maroon">S</font> 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="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span> ;<br/><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font> = 
 <a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> 
<b>by </b><i><a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="topalg_3.html#T6">TOPALG_3:6</a>, <a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">S1</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, , <a class="ref" href="tsep_1.html#T3">TSEP_1:3</a></i>
;<br/><a NAME="E18:40_2_3_1_1"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K0</font>
 <b>by </b><i><a class="ref" href="borsuk_3.html#T26">BORSUK_3:26</a></i>;<br/><b>hence </b><a NAME="E19:40_2_3_1_1"/>
<font color="Maroon">Fn</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/><a NAME="E20:40_2_3_1_1"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E21:40_2_3_1_1"/><b>then </b><i><font color="Green">E195</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#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E22:40_2_3_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Fn</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="toprealb.html#K12">CircleMap</a> 
 <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E23:40_2_3_1_1"/><b>then </b><i><font color="Green">E197</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fn</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><a NAME="E24:40_2_3_1_1"/>
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>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_1_2"><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="E1:40_2_3_1_1_2"/><i><font color="Green">E198</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"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Ft</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, , , , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, , <a class="ref" href="topalg_5.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E25:40_2_3_1_1"/>
<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn</font>
 <b>by </b><i>, , <a class="txt" href="topalg_5.html#E1:40"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><a NAME="E26:40_2_3_1_1"/><i><font color="Green">E199</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E27:40_2_3_1_1"/><i><font color="Green">E200</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="zfmisc_1.html#T124">ZFMISC_1:124</a></i>;<br/><i><font color="Green">E201</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="zfmisc_1.html#T124">ZFMISC_1:124</a></i>
;<br/><a NAME="E29:40_2_3_1_1"/>
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>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_1_4"><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="E1:40_2_3_1_1_4"/><i><font color="Green">E202</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"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:40_2_3_1_1_4"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:40_2_3_1_1_4"/><i><font color="Green">E204</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E4:40_2_3_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">Ft</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="txt" href="topalg_5.html#E4:40"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E30:40_2_3_1_1"/>
<font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_2"><b>suppose </b></a><a NAME="E1:40_2_3_1_2"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">N2</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Maroon">S</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">Fn</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E3:40_2_3_1_2"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">S</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"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E4:40_2_3_1_2"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font>
 <b>and </b><br/><a NAME="E5:40_2_3_1_2"/><i><font color="Green">E210</font></i>: 
<font color="Maroon">N2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E6:40_2_3_1_2"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">Fn</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E7:40_2_3_1_2"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn</font>
 <b>and </b><br/><a NAME="E8:40_2_3_1_2"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E7"><i><font color="Green">Lemma62</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">N2</font> = <font color="Maroon">N2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/><a NAME="E10:40_2_3_1_2"/><i><font color="Green">E214</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E11:40_2_3_1_2"/><i><font color="Green">E215</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/><b>set </b><font color="Maroon">SS</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>;<br/><b>consider </b><font color="Maroon">Ui</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="toprealb.html#K8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E13:40_2_3_1_2"/><i><font color="Green">E216</font></i>: 
<font color="Maroon">Ui</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UL</font>
 <b>and </b><br/><a NAME="E14:40_2_3_1_2"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ui</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, <a class="txt" href="topalg_5.html#E9"><i><font color="Green">Lemma64</font></i></a></i>;<br/><a NAME="E15:40_2_3_1_2"/><i><font color="Green">E218</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="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ui</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ui</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><b>consider </b><font color="Maroon">D</font> being  <a href="taxonom2.html#V4">mutually-disjoint</a> <a href="tops_2.html#V1">open</a> <a href="struct_0.html#NM3">Subset-Family</a> of <a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E17:40_2_3_1_2"/><i><font color="Green">E219</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">Ui</font>
 <b>and </b><br/><a NAME="E18:40_2_3_1_2"/><i><font color="Green">E220</font></i>: 
for <font color="Olive">d</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/>for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Olive">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ui</font></span>)</span>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Olive">d</font> holds <br/><font color="Olive">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, </i>;<br/><a NAME="E19:40_2_3_1_2"/><i><font color="Green">E221</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">N2</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E20:40_2_3_1_2"/><i><font color="Green">E222</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9"><i><font color="Green">Lemma64</font></i></a>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a></i>;<br/><a NAME="E21:40_2_3_1_2"/><b>then </b><i><font color="Green">E223</font></i>: 
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E22:40_2_3_1_2"/><b>then </b><i><font color="Green">E224</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then </b><i><font color="Green">E225</font></i>: <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="topalg_5.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E4:40"><i><font color="Green">E78</font></i></a>, , <a class="txt" href="topalg_5.html#E8:40"><i><font color="Green">E81</font></i></a>, <a class="txt" href="topalg_5.html#E1:40"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/><a NAME="E24:40_2_3_1_2"/><i><font color="Green">E226</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E80</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E25:40_2_3_1_2"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</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><a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, <a class="txt" href="topalg_5.html#E9"><i><font color="Green">Lemma64</font></i></a>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a></i>;<br/><a NAME="E26:40_2_3_1_2"/><b>then </b>
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</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"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E27:40_2_3_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E28:40_2_3_1_2"/><b>then </b><i><font color="Green">E228</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9:40"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/><a NAME="E29:40_2_3_1_2"/>
<font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">Ui</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E11:40"><i><font color="Green">E85</font></i></a>, , , <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><b>then consider </b><font color="Maroon">Uit</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E31:40_2_3_1_2"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Uit</font>
 <b>and </b><br/><a NAME="E32:40_2_3_1_2"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">Uit</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">Uit</font> = <font color="Maroon">Uit</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E12:40"><i><font color="Green">E86</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2"><i><font color="Green">E87</font></i></a></i>;<br/><a NAME="E34:40_2_3_1_2"/><i><font color="Green">E233</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E35:40_2_3_1_2"/><i><font color="Green">E234</font></i>: 
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E36:40_2_3_1_2"/><b>then </b><i><font color="Green">E235</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E37:40_2_3_1_2"/><i><font color="Green">E236</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E80</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>reconsider </b><font color="Maroon">Ti</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E80</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E39:40_2_3_1_2"/><i><font color="Green">E237</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ti</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ti</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E40:40_2_3_1_2"/><i><font color="Green">E238</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ti</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ti</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/><b>set </b><font color="Maroon">FnT</font> = <font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Ti</font></span><a href="borsuk_1.html#K6">:]</a></span>;<br/><a NAME="E41:40_2_3_1_2"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fn</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E4:40"><i><font color="Green">E78</font></i></a>, , <a class="txt" href="topalg_5.html#E1:40"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topalg_5.html#E7:40_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E42:40_2_3_1_2"/><b>then </b><i><font color="Green">E239</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Ti</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E43:40_2_3_1_2"/><i><font color="Green">E240</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Ti</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><b>reconsider </b><font color="Maroon">FnT</font> = <font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Ti</font></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ti</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E4:40"><i><font color="Green">E78</font></i></a>, <a class="txt" href="topalg_5.html#E13:40_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="topalg_5.html#E9:40_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="topalg_5.html#E12:40_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E45:40_2_3_1_2"/>
<font color="Maroon">N2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N2</font>
 ;<br/><b>then reconsider </b><font color="Maroon">N7</font> = <font color="Maroon">N2</font> 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"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><b>reconsider </b><font color="Maroon">Ti2</font> = <font color="Maroon">Ti</font> 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="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span> <b>by </b><i>, <a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E80</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E48:40_2_3_1_2"/><i><font color="Green">E241</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font> &amp; <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ti2</font> <a href="hidden.html#R1">=</a> <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ti</font> )
 <b>by </b><i><a class="ref" href="goboard9.html#T4">GOBOARD9:4</a></i>;<br/><a NAME="E49:40_2_3_1_2"/>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N7</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ti2</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N7</font>,<font color="Maroon">Ti2</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_3.html#T26">BORSUK_3:26</a></i>;<br/><a NAME="E50:40_2_3_1_2"/><b>then </b><i><font color="Green">E242</font></i>: 
<font color="Maroon">FnT</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E14:40_2"><i><font color="Green">E96</font></i></a>, , <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/><a NAME="E51:40_2_3_1_2"/>
<font color="Maroon">Uit</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/><a NAME="E52:40_2_3_1_2"/><b>then </b>
<font color="Maroon">FnT</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Uit</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/><b>then consider </b><font color="Maroon">SF</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ti</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E54:40_2_3_1_2"/><i><font color="Green">E243</font></i>: 
<font color="Maroon">FnT</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Uit</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">SF</font>
 <b>and </b><br/><a NAME="E55:40_2_3_1_2"/><i><font color="Green">E244</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> holds <br/> ex <font color="Olive">X1</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span> ex <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ti</font></span>)</span> st <br/>( <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y1</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T45">BORSUK_1:45</a></i>;<br/><a NAME="E56:40_2_3_1_2"/>
<font color="Maroon">FnT</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Uit</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E12:40"><i><font color="Green">E86</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E57:40_2_3_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">FnT</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Uit</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:40_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="topalg_5.html#E12:40_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">N5</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E59:40_2_3_1_2"/><i><font color="Green">E245</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">N5</font>
 <b>and </b><br/><a NAME="E60:40_2_3_1_2"/><i><font color="Green">E246</font></i>: 
<font color="Maroon">N5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">X1</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>, <font color="Maroon">Y1</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ti</font></span>)</span><b> such that </b><br/><a NAME="E62:40_2_3_1_2"/><i><font color="Green">E247</font></i>: 
<font color="Maroon">N5</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E63:40_2_3_1_2"/><i><font color="Green">E248</font></i>: 
<font color="Maroon">X1</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E64:40_2_3_1_2"/>
<font color="Maroon">Y1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">NY</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E66:40_2_3_1_2"/><i><font color="Green">E249</font></i>: 
<font color="Maroon">NY</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E67:40_2_3_1_2"/><i><font color="Green">E250</font></i>: 
<font color="Maroon">NY</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/><b>consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E69:40_2_3_1_2"/><i><font color="Green">E251</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>and </b><br/><a NAME="E70:40_2_3_1_2"/><i><font color="Green">E252</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 <b>and </b><br/><a NAME="E71:40_2_3_1_2"/><i><font color="Green">E253</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="txt" href="topalg_5.html#E15:40_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>set </b><font color="Maroon">N1</font> = <font color="Maroon">NY</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">N2</font>;<br/><a NAME="E72:40_2_3_1_2"/><i><font color="Green">E254</font></i>: 
<font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ti</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:40_2_3_1_2_2"/>
<font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ti</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9:40_2"><i><font color="Green">E92</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">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="E2:40_2_3_1_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ti</font>
 ;<br/>

<a NAME="E3:40_2_3_1_2_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:40_2_3_1_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 <b>by </b><i><a class="ref" href="topalg_5.html#T2" target="_self">Th2</a>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div><a NAME="E73:40_2_3_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E74:40_2_3_1_2"/><b>then </b><i><font color="Green">E255</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">NY</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E16:40_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="topalg_5.html#E17:40_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">N1</font> = <font color="Maroon">NY</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">N2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E76:40_2_3_1_2"/><i><font color="Green">E275</font></i>: 
<font color="Maroon">N1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E77:40_2_3_1_2"/><i><font color="Green">E276</font></i>: 
<font color="Maroon">Fn</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Uit</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:40_2_3_1_2_3"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fn</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E3:40_2_3_1_2_3"/><i><font color="Green">E277</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E4:40_2_3_1_2_3"/><i><font color="Green">E278</font></i>: 
<font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E5:40_2_3_1_2_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N5</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E16:40_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a>, , <a class="txt" href="topalg_5.html#E15:40_2"><i><font color="Green">E102</font></i></a>, </i>;<br/>
<a NAME="E6:40_2_3_1_2_3"/><b>then </b><i><font color="Green">E279</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">SF</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E7:40_2_3_1_2_3"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FnT</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E8:40_2_3_1_2_3"/><b>then </b>
<font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FnT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E9:40_2_3_1_2_3"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uit</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div><a NAME="E78:40_2_3_1_2"/><i><font color="Green">E280</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">N1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E79:40_2_3_1_2"/><i><font color="Green">E281</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E80:40_2_3_1_2"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E81:40_2_3_1_2"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E82:40_2_3_1_2"/><b>then </b><i><font color="Green">E282</font></i>: 
<font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/><b>set </b><font color="Maroon">f</font> = <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font>;<br/><a NAME="E83:40_2_3_1_2"/><i><font color="Green">E283</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">Uit</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Uit</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E84:40_2_3_1_2"/><i><font color="Green">E284</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Uit</font>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E85:40_2_3_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ui</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:40_2_3_1_2_4"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_2_3_1_2_4"/><i><font color="Green">E285</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font></span>)</span>
 <b>and </b><br/><a NAME="E4:40_2_3_1_2_4"/><i><font color="Green">E286</font></i>: 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:40_2_3_1_2_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font>
 
<b>by </b><i>, , <a class="txt" href="topalg_5.html#E2:40_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:40_2_3_1_2_4"/><b>then </b>
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ui</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<b>hence </b><a NAME="E7:40_2_3_1_2_4"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ui</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">f</font> = <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">Uit</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ui</font></span>)</span> <b>by </b><i>, , <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E87:40_2_3_1_2"/><i><font color="Green">E287</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, <a class="txt" href="topalg_5.html#E2:40_2"><i><font color="Green">E87</font></i></a></i>;<br/><a NAME="E88:40_2_3_1_2"/>
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, <a class="txt" href="topalg_5.html#E9"><i><font color="Green">Lemma64</font></i></a>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">SS</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a>, <a class="txt" href="topalg_5.html#E5:40"><i><font color="Green">E79</font></i></a>, <a class="txt" href="topalg_5.html#E5"><i><font color="Green">Lemma59</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><b>take </b>
<font color="Maroon">N1</font>
;<br/><b>take </b><font color="Maroon">S1</font> = <font color="Maroon">S</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">SS</font>;<br/><b>set </b><font color="Maroon">Fni1</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">f</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><a NAME="E91:40_2_3_1_2"/><i><font color="Green">E290</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E92:40_2_3_1_2"/>
<font color="Maroon">f</font> <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="E93:40_2_3_1_2"/><b>then </b><i><font color="Green">E291</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </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="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ui</font></span>)</span>
 <b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E94:40_2_3_1_2"/><i><font color="Green">E292</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</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="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ui</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E95:40_2_3_1_2"/><i><font color="Green">E293</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="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E96:40_2_3_1_2"/><b>then </b><i><font color="Green">E294</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">ff</font> <a href="funct_1.html#K2">"</a> 
 <b>by </b><i>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/><a NAME="E97:40_2_3_1_2"/><i><font color="Green">E295</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">SS</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">SS</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E98:40_2_3_1_2"/><i><font color="Green">E296</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E99:40_2_3_1_2"/><i><font color="Green">E297</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#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E100:40_2_3_1_2"/><i><font color="Green">E298</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#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:40_2_3_1_2_5"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_2_3_1_2_5"/><i><font color="Green">E299</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:40_2_3_1_2_5"/><i><font color="Green">E300</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:40_2_3_1_2_5"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E6:40_2_3_1_2_5"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E7:40_2_3_1_2_5"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:40_2_3_1_2_5"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ui</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:40_2_3_1_2_5"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>


</div><b>end;</b></div><a NAME="E101:40_2_3_1_2"/><b>then </b><i><font color="Green">E301</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">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><a NAME="E102:40_2_3_1_2"/><i><font color="Green">E302</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">SS</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/><a NAME="E103:40_2_3_1_2"/><i><font color="Green">E303</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S1</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i>, <a class="ref" href="topalg_5.html#T4" target="_self">Th4</a>, <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/><a NAME="E104:40_2_3_1_2"/><i><font color="Green">E304</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/><a NAME="E105:40_2_3_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">Uit</font></span>)</span> is   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/><a NAME="E106:40_2_3_1_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3">R^1</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">Fni1</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">SS</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  <b>by </b><i>, , , <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><b>set </b><font color="Maroon">Fn2</font> = <font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span>;<br/><b>set </b><font color="Maroon">Fn1</font> = <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">Fni1</font>;<br/><a NAME="E108:40_2_3_1_2"/><i><font color="Green">E307</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T120">ZFMISC_1:120</a></i>;<br/><a NAME="E109:40_2_3_1_2"/><i><font color="Green">E308</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E110:40_2_3_1_2"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:40"><i><font color="Green">E78</font></i></a>, , <a class="txt" href="topalg_5.html#E1:40"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E111:40_2_3_1_2"/><b>then </b><i><font color="Green">E309</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">Fni1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E112:40_2_3_1_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">Fni1</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Fni1</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/><a NAME="E113:40_2_3_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">Fni1</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3">R^1</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">Fn1</font> = <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">Fni1</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S1</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topalg_5.html#T8" target="_self">Th8</a>, , <a class="ref" href="topalg_5.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><b>take </b>
<font color="Maroon">Fn1</font>
;<br/><b>thus </b><a NAME="E115:40_2_3_1_2"/><i><font color="Green">E310</font></i>: 
<font color="Maroon">S1</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"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="txt" href="topalg_5.html#E5:40"><i><font color="Green">E79</font></i></a>, <a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a>, <a class="ref" href="heine.html#T2">HEINE:2</a></i>;<br/><b>thus </b><a NAME="E116:40_2_3_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b><a NAME="E117:40_2_3_1_2"/>
<font color="Maroon">N1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E118:40_2_3_1_2"/><i><font color="Green">E311</font></i>: 
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E119:40_2_3_1_2"/><i><font color="Green">E312</font></i>: 
<font color="Maroon">SS</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E120:40_2_3_1_2"/><i><font color="Green">E313</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i>, , <a class="txt" href="topalg_5.html#E4:40"><i><font color="Green">E78</font></i></a>, , <a class="txt" href="topalg_5.html#E1:40"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E121:40_2_3_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3">R^1</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">Fn2</font> = <font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E123:40_2_3_1_2"/><i><font color="Green">E314</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S1</font></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#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S1</font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">F1</font> =  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span>, <font color="Maroon">F2</font> =  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">SS</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S1</font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a>, <a class="ref" href="topalg_5.html#T4" target="_self">Th4</a>, <a class="ref" href="topalg_5.html#T10" target="_self">Th10</a>, </i>;<br/><a NAME="E125:40_2_3_1_2"/><i><font color="Green">E317</font></i>: 
<a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S1</font>
 <b>by </b><i>, <a class="ref" href="topalg_5.html#T4" target="_self">Th4</a>, <a class="ref" href="topalg_5.html#T10" target="_self">Th10</a>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/><a NAME="E126:40_2_3_1_2"/><i><font color="Green">E318</font></i>: 
<a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">SS</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S1</font>
 <b>by </b><i>, <a class="ref" href="topalg_5.html#T4" target="_self">Th4</a>, , <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/><b>reconsider </b><font color="Maroon">hS</font> = <font color="Maroon">F1</font>, <font color="Maroon">hSS</font> = <font color="Maroon">F2</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E128:40_2_3_1_2"/>
<font color="Maroon">hS</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, , <a class="ref" href="borsuk_4.html#T48">BORSUK_4:48</a></i>;<br/><a NAME="E129:40_2_3_1_2"/><b>then </b><i><font color="Green">E321</font></i>: 
<font color="Maroon">F1</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tsep_1.html#T8">TSEP_1:8</a></i>;<br/><a NAME="E130:40_2_3_1_2"/>
<font color="Maroon">hSS</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="borsuk_4.html#T48">BORSUK_4:48</a></i>;<br/><a NAME="E131:40_2_3_1_2"/><b>then </b><i><font color="Green">E322</font></i>: 
<font color="Maroon">F2</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tsep_1.html#T8">TSEP_1:8</a></i>;<br/><b>reconsider </b><font color="Maroon">K0</font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i>, <a class="txt" href="topalg_5.html#E1:40"><i><font color="Green">E72</font></i></a>, , <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><b>reconsider </b><font color="Maroon">aN1</font> = <font color="Maroon">N1</font> 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"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span> <b>by </b><i>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E134:40_2_3_1_2"/>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 ;<br/><b>then reconsider </b><font color="Maroon">aS</font> = <font color="Maroon">S</font> 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="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K0</font> = 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">aN1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">aS</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_3.html#T26">BORSUK_3:26</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">aS</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="goboard9.html#T4">GOBOARD9:4</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="goboard9.html#T4">GOBOARD9:4</a></i>
;<br/><a NAME="E137:40_2_3_1_2"/><b>then </b><i><font color="Green">E325</font></i>: 
<font color="Maroon">Fn2</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/><b>reconsider </b><font color="Maroon">fF</font> = <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">SS</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ui</font></span>)</span> <b>by </b><i>, , <a class="ref" href="topalg_5.html#T5" target="_self">Th5</a>, , <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><b>reconsider </b><font color="Maroon">gF</font> = <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">SS</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <b>by </b><i>, <a class="ref" href="topalg_5.html#T5" target="_self">Th5</a>, , <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E140:40_2_3_1_2"/>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">SS</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_3.html#T26">BORSUK_3:26</a></i>;<br/><a NAME="E141:40_2_3_1_2"/><b>then </b>
<font color="Maroon">gF</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/><a NAME="E142:40_2_3_1_2"/><b>then </b><i><font color="Green">E328</font></i>: 
<font color="Maroon">fF</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T9">TOPMETR:9</a></i>;<br/><a NAME="E143:40_2_3_1_2"/>
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  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="E144:40_2_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</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="topalg_5.html#T13" target="_self">Th13</a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/><a NAME="E145:40_2_3_1_2"/><b>then </b><i><font color="Green">E329</font></i>: 
<font color="Maroon">Fni1</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="E146:40_2_3_1_2"/>
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">SS</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> holds <br/><font color="Maroon">Fn2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fni1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_2_7"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_2_3_1_2_7"/><i><font color="Green">E330</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">SS</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:40_2_3_1_2_7"/><i><font color="Green">E331</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">SS</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T122">ZFMISC_1:122</a></i>;<br/>
<a NAME="E3:40_2_3_1_2_7"/><i><font color="Green">E332</font></i>: 
<font color="Maroon">S</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">SS</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="txt" href="topalg_5.html#E5:40"><i><font color="Green">E79</font></i></a>, <a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a>, <a class="ref" href="topmetr2.html#T1">TOPMETR2:1</a></i>;<br/>
<a NAME="E4:40_2_3_1_2_7"/><i><font color="Green">E333</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">SS</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E5:40_2_3_1_2_7"/><i><font color="Green">E334</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">SS</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E6:40_2_3_1_2_7"/><i><font color="Green">E335</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T16" target="_self">Th16</a>, <a class="ref" href="topalg_5.html#T14" target="_self">Th14</a>, , <a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="txt" href="topalg_5.html#E5:40"><i><font color="Green">E79</font></i></a>, <a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a>, <a class="ref" href="topmetr2.html#T1">TOPMETR2:1</a>, </i>;<br/>
<b>then consider </b><font color="Maroon">p1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">N1</font>, <font color="Maroon">p2</font> being    <a href="subset_1.html#M2">Element</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K6">}</a></span><b> such that </b><br/><a NAME="E8:40_2_3_1_2_7"/><i><font color="Green">E337</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E9:40_2_3_1_2_7"/><i><font color="Green">E338</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">N2</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E10:40_2_3_1_2_7"/><i><font color="Green">E339</font></i>: 
<font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E11:40_2_3_1_2_7"/><i><font color="Green">E340</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T17" target="_self">Th17</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:40_2_3_1_2_7"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 
;<br/>
<a NAME="E13:40_2_3_1_2_7"/><b>then </b><i><font color="Green">E341</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E14:40_2_3_1_2_7"/><b>then </b><i><font color="Green">E342</font></i>: 
<font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fn</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="txt" href="topalg_5.html#E4:40"><i><font color="Green">E78</font></i></a>, , <a class="txt" href="topalg_5.html#E1:40"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> = 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="topalg_5.html#D1" target="_self">Def1</a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="topalg_5.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="topalg_5.html#T1" target="_self">Th1</a>, <a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green">E88</font></i></a>, , <a class="txt" href="topalg_5.html#E1:40"><i><font color="Green">E72</font></i></a>, , <a class="ref" href="topalg_5.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><font color="Maroon">Fn2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="topalg_5.html#D2" target="_self">Def2</a>, , <a class="ref" href="topalg_5.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
<br/>.= 
<font color="Maroon">Fni1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="topalg_5.html#D1" target="_self">Def1</a>, , , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><br/>


</div><b>end;</b></div><a NAME="E147:40_2_3_1_2"/><b>then </b>
 ex <font color="Olive">h</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S1</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  st <br/>( <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn2</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">Fni1</font> &amp; <font color="Olive">h</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i>, , , <a class="txt" href="topalg_5.html#E24"><i><font color="Green">Lemma146</font></i></a>, <a class="ref" href="topalg_5.html#T12" target="_self">Th12</a>, , , , <a class="ref" href="topalg_5.html#T4" target="_self">Th4</a>, <a class="ref" href="topalg_5.html#T11" target="_self">Th11</a>, <a class="ref" href="topalg_3.html#T20">TOPALG_3:20</a></i>;<br/><b>hence </b><a NAME="E148:40_2_3_1_2"/>
<font color="Maroon">Fn1</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/><a NAME="E149:40_2_3_1_2"/><i><font color="Green">E344</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E150:40_2_3_1_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Fn1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="toprealb.html#K12">CircleMap</a> 
 <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E151:40_2_3_1_2"/><b>then </b><i><font color="Green">E345</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fn1</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><a NAME="E152:40_2_3_1_2"/><i><font color="Green">E346</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fn1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, <a class="ref" href="topalg_5.html#T8" target="_self">Th8</a>, <a class="ref" href="topalg_5.html#T14" target="_self">Th14</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E153:40_2_3_1_2"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_2_8"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_2_3_1_2_8"/><i><font color="Green">E347</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_2_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_3_1_2_8_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fni1</font> or  not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fni1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_2_8_1_1"><b>suppose </b></a><a NAME="E1:40_2_3_1_2_8_1_1"/><i><font color="Green">E348</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fni1</font>
 ;<br/><div class="add"><a NAME="E2:40_2_3_1_2_8_1_1"/><b>then </b><i><font color="Green">E349</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E3:40_2_3_1_2_8_1_1"/><i><font color="Green">E350</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E4:40_2_3_1_2_8_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E5:40_2_3_1_2_8_1_1"/><b>then </b><i><font color="Green">E351</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i/>;<br/><a NAME="E6:40_2_3_1_2_8_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">ff</font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span>
 <b>by </b><i>, , , , , , <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E7:40_2_3_1_2_8_1_1"/><b>then </b><i><font color="Green">E352</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ff</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T182">RELAT_1:182</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="topalg_5.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fni1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <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"><font color="Maroon">ff</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, , , , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">ff</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="topalg_5.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">ff</font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<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="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">ff</font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, , , , <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">a</font>
<b>by </b><i><a class="ref" href="topalg_5.html#T3" target="_self">Th3</a>, , , , <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, <a class="ref" href="topalg_3.html#T2">TOPALG_3:2</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_2_8_1_2"><b>suppose </b></a><a NAME="E1:40_2_3_1_2_8_1_2"/><i><font color="Green">E353</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fni1</font>
 ;<br/><div class="add"><a NAME="E2:40_2_3_1_2_8_1_2"/><b>then </b><i><font color="Green">E354</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="ref" href="topalg_5.html#T8" target="_self">Th8</a>, , , <a class="ref" href="topalg_5.html#D1" target="_self">Def1</a>, <a class="ref" href="topalg_5.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="topalg_5.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E4:40"><i><font color="Green">E78</font></i></a>, , <a class="txt" href="topalg_5.html#E1:40"><i><font color="Green">E72</font></i></a>, , , <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="topalg_5.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E154:40_2_3_1_2"/>
<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font>
 <b>by </b><i><a class="ref" href="topalg_5.html#T16" target="_self">Th16</a>, <a class="ref" href="topalg_5.html#T17" target="_self">Th17</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E155:40_2_3_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</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><a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a></i>;<br/><a NAME="E156:40_2_3_1_2"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E157:40_2_3_1_2"/><b>then </b><i><font color="Green">E355</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S1</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E158:40_2_3_1_2"/><i><font color="Green">E356</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">Fn1</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><b>then </b><i><font color="Green">E357</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">N1</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="subset_1.html#K9">/\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, <a class="ref" href="topalg_5.html#T8" target="_self">Th8</a>, <a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="subset_1.html#K9">/\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="topalg_5.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><i><font color="Green">E358</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">Ft</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">N1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,<span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><a NAME="E161:40_2_3_1_2"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_2_11"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_2_3_1_2_11"/><i><font color="Green">E359</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:40_2_3_1_2_11"/><b>then </b><i><font color="Green">E360</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">a1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">a2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:40_2_3_1_2_11"/>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E5:40_2_3_1_2_11"/><i><font color="Green">E361</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E6:40_2_3_1_2_11"/><i><font color="Green">E362</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E7:40_2_3_1_2_11"/><i><font color="Green">E363</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:40_2_3_1_2_11"/>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E5:40"><i><font color="Green">E79</font></i></a>, <a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E9:40_2_3_1_2_11"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E10:40_2_3_1_2_11"/><b>then </b><i><font color="Green">E364</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E11:40_2_3_1_2_11"/><b>then </b><i><font color="Green">E365</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:40_2_3_1_2_11"/><i><font color="Green">E366</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E13:40_2_3_1_2_11"/><b>then </b><i><font color="Green">E367</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E14:40_2_3_1_2_11"/><i><font color="Green">E368</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_2_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_3_1_2_11_1"/>
(  not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fni1</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fni1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_2_11_1_1"><b>suppose </b></a><a NAME="E1:40_2_3_1_2_11_1_1"/><i><font color="Green">E369</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fni1</font>
 ;<br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="topalg_5.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E34"><i><font color="Green">Lemma164</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="topalg_5.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">Ft</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_2_11_1_2"><b>suppose </b></a><a NAME="E1:40_2_3_1_2_11_1_2"/><i><font color="Green">E370</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fni1</font>
 ;<br/><div class="add"><a NAME="E2:40_2_3_1_2_11_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><b>then consider </b><font color="Maroon">b1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:40_2_3_1_2_11_1_2"/><i><font color="Green">E373</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 <b>and </b><br/><a NAME="E5:40_2_3_1_2_11_1_2"/><i><font color="Green">E374</font></i>: 
<font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SS</font>
 <b>and </b><br/><a NAME="E6:40_2_3_1_2_11_1_2"/><i><font color="Green">E375</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>set </b><font color="Maroon">e</font> = <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>;<br/><i><font color="Green">E376</font></i>: <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">Ft</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="topalg_5.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><a NAME="E8:40_2_3_1_2_11_1_2"/>
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font>
 <b>by </b><i><a class="ref" href="topalg_5.html#D3" target="_self">Def3</a>, <a class="ref" href="topalg_5.html#T20" target="_self">Th20</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E9:40_2_3_1_2_11_1_2"/><b>then </b><i><font color="Green">E377</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E5:40"><i><font color="Green">E79</font></i></a>, , <a class="ref" href="topalg_5.html#D4" target="_self">Def4</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E10:40_2_3_1_2_11_1_2"/><i><font color="Green">E378</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="topalg_5.html#D3" target="_self">Def3</a>, <a class="ref" href="topalg_5.html#T20" target="_self">Th20</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E11:40_2_3_1_2_11_1_2"/><b>then </b><i><font color="Green">E379</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E35"><i><font color="Green">Lemma165</font></i></a>, <a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E80</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E12:40_2_3_1_2_11_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E35"><i><font color="Green">Lemma165</font></i></a>, <a class="txt" href="topalg_5.html#E39"><i><font color="Green">Lemma170</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E13:40_2_3_1_2_11_1_2"/><b>then </b><i><font color="Green">E380</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fn</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:40"><i><font color="Green">E78</font></i></a>, <a class="txt" href="topalg_5.html#E38"><i><font color="Green">Lemma168</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E1:40"><i><font color="Green">E72</font></i></a>, , , <a class="ref" href="topalg_5.html#D3" target="_self">Def3</a>, , <a class="txt" href="topalg_5.html#E40"><i><font color="Green">Lemma171</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><i><font color="Green">E381</font></i>: <font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span> = 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E41"><i><font color="Green">Lemma172</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Ft</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="topalg_5.html#D2" target="_self">Def2</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">a</font>
<b>by </b><i>, <a class="ref" href="topalg_5.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="topalg_5.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">Fni1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E34"><i><font color="Green">Lemma164</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">SS</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="txt" href="topalg_5.html#E34"><i><font color="Green">Lemma164</font></i></a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ff</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="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="topalg_5.html#E34"><i><font color="Green">Lemma164</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="txt" href="topalg_5.html#E41"><i><font color="Green">Lemma172</font></i></a>, , , <a class="ref" href="topalg_5.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E162:40_2_3_1_2"/>
<font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E18:40_2"/><i><font color="Green">E382</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>consider </b><font color="Maroon">N2</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Maroon">S</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">Fn1</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E20:40_2"/><i><font color="Green">E383</font></i>: 
<font color="Maroon">S</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"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">TT</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E21:40_2"/><i><font color="Green">E384</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font>
 <b>and </b><br/><a NAME="E22:40_2"/><i><font color="Green">E385</font></i>: 
<font color="Maroon">N2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E23:40_2"/><i><font color="Green">E386</font></i>: 
<font color="Maroon">Fn1</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E24:40_2"/><i><font color="Green">E387</font></i>: 
<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">S</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font>
 <b>and </b><br/><a NAME="E25:40_2"/><i><font color="Green">E388</font></i>: 
<font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E7"><i><font color="Green">Lemma62</font></i></a>, <a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma57</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>take </b>
<font color="Maroon">t</font>
;<br/>

<b>take </b>
<font color="Maroon">N2</font>
;<br/>

<a NAME="E27:40_2"/><i><font color="Green">E389</font></i>: 
<a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, , , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="tsep_1.html#T3">TSEP_1:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Fn1</font> = <font color="Maroon">Fn1</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  ;<br/>
<b>take </b>
<font color="Maroon">Fn1</font>
;<br/>

<b>thus </b><a NAME="E29:40_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">t</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font> &amp; <font color="Maroon">Fn1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font> &amp; <font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i>, <a class="txt" href="topalg_5.html#E9"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="topalg_5.html#T1" target="_self">Th1</a>, <a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, , , , , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>

<b>let </b><font color="Maroon">H</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E30:40_2"/><i><font color="Green">E390</font></i>: 
<font color="Maroon">H</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E31:40_2"/><i><font color="Green">E391</font></i>: 
<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E32:40_2"/><i><font color="Green">E392</font></i>: 
<font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E33:40_2"/><i><font color="Green">E393</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E34:40_2"/><i><font color="Green">E394</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E35:40_2"/><i><font color="Green">E395</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">N2</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E36:40_2"/><i><font color="Green">E396</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fn1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font> implies  ex <font color="Olive">Z</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Olive">Z</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"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Olive">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Olive">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> ) );<br/>
<a NAME="E37:40_2"/><i><font color="Green">E397</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_3.html#T26">FINSEQ_3:26</a></i>;<br/>
<a NAME="E38:40_2"/><i><font color="Green">E398</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4"><b>proof </b></a><div class="add">

<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><b>that </b><br/><a NAME="E1:40_2_4"/><i><font color="Green">E399</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">i</font>]
 <b>and </b><br/><a NAME="E2:40_2_4"/><i><font color="Green">E400</font></i>: 
<font color="Maroon">i</font> <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">TT</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_4_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font> )
 </span><b>by </b><i><a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E80</font></i></a>, <a class="ref" href="topreala.html#T23">TOPREALA:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_4_1_1"><b>suppose </b></a><a NAME="E1:40_2_4_1_1"/><i><font color="Green">E401</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">Z</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>;<br/><a NAME="E2:40_2_4_1_1"/><i><font color="Green">E402</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="txt" href="topalg_5.html#E8:40"><i><font color="Green">E81</font></i></a>, <a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>;<br/><b>reconsider </b><font color="Maroon">Z</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i>, , <a class="txt" href="topalg_5.html#E8:40"><i><font color="Green">E81</font></i></a>, <a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>;<br/><b>take </b>
<font color="Maroon">Z</font>
;<br/><b>thus </b><a NAME="E4:40_2_4_1_1"/>
<font color="Maroon">Z</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"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><a NAME="E5:40_2_4_1_1"/><i><font color="Green">E403</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/><a NAME="E6:40_2_4_1_1"/><b>then </b><i><font color="Green">E404</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, , <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E7:40_2_4_1_1"/><i><font color="Green">E405</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, , <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E9:40"><i><font color="Green">E83</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E8:40_2_4_1_1"/>
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>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4_1_1_1"><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="E1:40_2_4_1_1_1"/><i><font color="Green">E406</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"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:40_2_4_1_1_1"/><i><font color="Green">E407</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E12:40"><i><font color="Green">E86</font></i></a>, <a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="txt" href="topalg_5.html#E12:40"><i><font color="Green">E86</font></i></a>, <a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="txt" href="topalg_5.html#E1:40"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2"><i><font color="Green">E87</font></i></a>, <a class="txt" href="topalg_5.html#E12:40"><i><font color="Green">E86</font></i></a>, <a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a>, <a class="txt" href="topalg_5.html#E12:40"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:40_2_4_1_1"/>
<font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a>, <a class="txt" href="topalg_5.html#E11:40"><i><font color="Green">E85</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_4_1_2"><b>suppose </b></a><a NAME="E1:40_2_4_1_2"/><i><font color="Green">E408</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">Z</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E3:40_2_4_1_2"/><i><font color="Green">E409</font></i>: 
<font color="Maroon">Z</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"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E4:40_2_4_1_2"/><i><font color="Green">E410</font></i>: 
<font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E5:40"><i><font color="Green">E79</font></i></a></i>;<br/><b>set </b><font color="Maroon">ZZ</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>;<br/><a NAME="E5:40_2_4_1_2"/><i><font color="Green">E412</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E8:40"><i><font color="Green">E81</font></i></a>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a></i>;<br/><a NAME="E6:40_2_4_1_2"/><i><font color="Green">E413</font></i>: 
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E80</font></i></a>, <a class="txt" href="topalg_5.html#E8:40"><i><font color="Green">E81</font></i></a>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">ZZ</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a>, <a class="txt" href="topalg_5.html#E5"><i><font color="Green">Lemma59</font></i></a></i>;<br/><b>take </b><font color="Maroon">Z1</font> = <font color="Maroon">Z</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">ZZ</font>;<br/><a NAME="E8:40_2_4_1_2"/><i><font color="Green">E414</font></i>: 
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</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><a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E80</font></i></a>, <a class="txt" href="topalg_5.html#E8:40"><i><font color="Green">E81</font></i></a>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a></i>;<br/><b>hence </b><a NAME="E9:40_2_4_1_2"/>
<font color="Maroon">Z1</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"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a>, <a class="ref" href="heine.html#T2">HEINE:2</a></i>;<br/><a NAME="E10:40_2_4_1_2"/><i><font color="Green">E415</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/><a NAME="E11:40_2_4_1_2"/><b>then </b><i><font color="Green">E416</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, , <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E12:40_2_4_1_2"/><i><font color="Green">E417</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, , <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E13:40_2_4_1_2"/>
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>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4_1_2_1"><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="E1:40_2_4_1_2_1"/><i><font color="Green">E418</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"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_2_4_1_2_1"/><i><font color="Green">E420</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font>
 <b>and </b><br/><a NAME="E4:40_2_4_1_2_1"/><i><font color="Green">E421</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font>
 <b>and </b><br/><a NAME="E5:40_2_4_1_2_1"/><i><font color="Green">E422</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xy</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="txt" href="topalg_5.html#E7:40_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:40_2_4_1_2_1"/><i><font color="Green">E424</font></i>: 
<font color="Maroon">xy</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N2</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E7:40_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E8:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E425</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xZZ</font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="txt" href="topalg_5.html#E12:40_2"><i><font color="Green">E94</font></i></a>, , <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>consider </b><font color="Maroon">Ui</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="toprealb.html#K8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E11:40_2_4_1_2_1"/><i><font color="Green">E427</font></i>: 
<font color="Maroon">Ui</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UL</font>
 <b>and </b><br/><a NAME="E12:40_2_4_1_2_1"/><i><font color="Green">E428</font></i>: 
<font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ui</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E80</font></i></a>, <a class="txt" href="topalg_5.html#E8:40"><i><font color="Green">E81</font></i></a></i>;<br/>
<a NAME="E13:40_2_4_1_2_1"/><i><font color="Green">E429</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="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ui</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ui</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>consider </b><font color="Maroon">D</font> being  <a href="taxonom2.html#V4">mutually-disjoint</a> <a href="tops_2.html#V1">open</a> <a href="struct_0.html#NM3">Subset-Family</a> of <a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E15:40_2_4_1_2_1"/><i><font color="Green">E430</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">Ui</font>
 <b>and </b><br/><a NAME="E16:40_2_4_1_2_1"/><i><font color="Green">E431</font></i>: 
for <font color="Olive">d</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/>for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Olive">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ui</font></span>)</span>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Olive">d</font> holds <br/><font color="Olive">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E14:40_2"><i><font color="Green">E96</font></i></a>, </i>;<br/>
<a NAME="E17:40_2_4_1_2_1"/>
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, <a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E18:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E432</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E7:40_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E19:40_2_4_1_2_1"/><i><font color="Green">E433</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<b>then </b><i><font color="Green">E434</font></i>: <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, , , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E15:40_2"><i><font color="Green">E102</font></i></a>, , , <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E21:40_2_4_1_2_1"/><i><font color="Green">E435</font></i>: 
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E12:40"><i><font color="Green">E86</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E22:40_2_4_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E7:40_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E23:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E436</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E24:40_2_4_1_2_1"/><b>then </b>
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">Ui</font>
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>then consider </b><font color="Maroon">Uit</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E26:40_2_4_1_2_1"/><i><font color="Green">E437</font></i>: 
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Uit</font>
 <b>and </b><br/><a NAME="E27:40_2_4_1_2_1"/><i><font color="Green">E438</font></i>: 
<font color="Maroon">Uit</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Uit</font> = <font color="Maroon">Uit</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E17:40_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="topalg_5.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E29:40_2_4_1_2_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E80</font></i></a>, <a class="txt" href="topalg_5.html#E8:40"><i><font color="Green">E81</font></i></a>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a></i>;<br/>
<a NAME="E30:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E439</font></i>: 
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E31:40_2_4_1_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</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><a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E80</font></i></a>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a></i>;<br/>
<a NAME="E32:40_2_4_1_2_1"/><b>then </b>
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40"><i><font color="Green">E85</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E33:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E440</font></i>: 
<font color="Maroon">ZZ</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, <a class="txt" href="topalg_5.html#E12:40"><i><font color="Green">E86</font></i></a>, <a class="ref" href="borsuk_4.html#T49">BORSUK_4:49</a></i>;<br/>
<a NAME="E34:40_2_4_1_2_1"/>
<font color="Maroon">xy</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E7:40_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="connsp_1.html#T29">CONNSP_1:29</a></i>;<br/>
<a NAME="E35:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E441</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, <a class="ref" href="topalg_3.html#T17">TOPALG_3:17</a></i>;<br/>
<a NAME="E36:40_2_4_1_2_1"/>
 <a href="topmetr.html#K5">I[01]</a>  is    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#T2">TSEP_1:2</a></i>;<br/>
<a NAME="E37:40_2_4_1_2_1"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T25">BORSUK_3:25</a></i>;<br/>
<a NAME="E38:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E442</font></i>: 
<font color="Maroon">xZZ</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>
<a NAME="E39:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E443</font></i>: 
<font color="Maroon">Fn1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, <a class="ref" href="topalg_5.html#T1" target="_self">Th1</a>, <a class="ref" href="tops_2.html#T75">TOPS_2:75</a></i>;<br/>
<a NAME="E40:40_2_4_1_2_1"/><i><font color="Green">E444</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xy</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E41:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E445</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">xZZ</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E42:40_2_4_1_2_1"/><b>then </b>
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Fn1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E43:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E446</font></i>: 
<font color="Maroon">Uit</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Fn1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E17:40_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E44:40_2_4_1_2_1"/>
<font color="Maroon">D</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">Fn1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a>,<a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_2_4_1_2_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fn1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E3:40_2_4_1_2_1_2"/><i><font color="Green">E447</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xZZ</font>
 <b>and </b><br/><a NAME="E4:40_2_4_1_2_1_2"/><i><font color="Green">E448</font></i>: 
<font color="Maroon">Fn1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<i><font color="Green">E449</font></i>: <a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, , , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E6:40_2_4_1_2_1_2"/>
<font color="Maroon">xy</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="txt" href="topalg_5.html#E7:40_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:40_2_4_1_2_1_2"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E8:40_2_4_1_2_1_2"/><b>then </b><i><font color="Green">E450</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:40_2_4_1_2_1_2"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E10:40_2_4_1_2_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E11:40_2_4_1_2_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font>
 <b>by </b><i>, , , , , <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E45:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E451</font></i>: 
<font color="Maroon">Fn1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Uit</font>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T2" target="_self">Th2</a>, , , <a class="ref" href="topalg_3.html#T13">TOPALG_3:13</a></i>;<br/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="txt" href="topalg_5.html#E15:40_2"><i><font color="Green">E102</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E15:40_2"><i><font color="Green">E102</font></i></a>, , <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E47:40_2_4_1_2_1"/><b>then </b>
<font color="Maroon">H</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">Ui</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E16:40_2"><i><font color="Green">E105</font></i></a>, , <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>then consider </b><font color="Maroon">Uith</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E49:40_2_4_1_2_1"/><i><font color="Green">E453</font></i>: 
<font color="Maroon">H</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Uith</font>
 <b>and </b><br/><a NAME="E50:40_2_4_1_2_1"/><i><font color="Green">E454</font></i>: 
<font color="Maroon">Uith</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Uith</font> = <font color="Maroon">Uith</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E52:40_2_4_1_2_1"/><i><font color="Green">E455</font></i>: 
<font color="Maroon">H</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, , <a class="ref" href="tops_2.html#T75">TOPS_2:75</a></i>;<br/>
<a NAME="E53:40_2_4_1_2_1"/>
<font color="Maroon">H</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E54:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E456</font></i>: 
<font color="Maroon">Uith</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">H</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E55:40_2_4_1_2_1"/>
<font color="Maroon">D</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">H</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4_1_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a>,<a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_2_4_1_2_1_4"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E3:40_2_4_1_2_1_4"/><i><font color="Green">E457</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xZZ</font>
 <b>and </b><br/><a NAME="E4:40_2_4_1_2_1_4"/><i><font color="Green">E458</font></i>: 
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<i><font color="Green">E459</font></i>: <a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, , <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E6:40_2_4_1_2_1_4"/>
<font color="Maroon">xy</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="txt" href="topalg_5.html#E7:40_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:40_2_4_1_2_1_4"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E8:40_2_4_1_2_1_4"/><b>then </b><i><font color="Green">E460</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:40_2_4_1_2_1_4"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E10:40_2_4_1_2_1_4"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E11:40_2_4_1_2_1_4"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="topalg_5.html#T3" target="_self">Th3</a>, , , , <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E56:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E461</font></i>: 
<font color="Maroon">H</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Uith</font>
 
<b>by </b><i>, , , <a class="ref" href="topalg_3.html#T13">TOPALG_3:13</a></i>;<br/>
<a NAME="E57:40_2_4_1_2_1"/><i><font color="Green">E462</font></i>: 
<font color="Maroon">H</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="txt" href="topalg_5.html#E9:40"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E59:40_2_4_1_2_1"/><b>then </b>
<font color="Maroon">Uit</font> <a href="subset_1.html#R2">meets</a> <font color="Maroon">Uith</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E17:40_2"><i><font color="Green">E106</font></i></a>, , <a class="ref" href="topalg_5.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E60:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E463</font></i>: 
<font color="Maroon">Uit</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Uith</font>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T2" target="_self">Th2</a>, , <a class="ref" href="taxonom2.html#D5">TAXONOM2:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">f</font> = <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font>;<br/>
<a NAME="E61:40_2_4_1_2_1"/><i><font color="Green">E464</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">Uit</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Uit</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E62:40_2_4_1_2_1"/><i><font color="Green">E465</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Uit</font>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E63:40_2_4_1_2_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ui</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4_1_2_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:40_2_4_1_2_1_6"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_2_4_1_2_1_6"/><i><font color="Green">E466</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font></span>)</span>
 <b>and </b><br/><a NAME="E4:40_2_4_1_2_1_6"/><i><font color="Green">E467</font></i>: 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:40_2_4_1_2_1_6"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="topalg_5.html#T4" target="_self">Th4</a>, <a class="ref" href="topalg_5.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:40_2_4_1_2_1_6"/><b>then </b>
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ui</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<b>hence </b><a NAME="E7:40_2_4_1_2_1_6"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ui</font>
 <b>by </b><i>, <a class="ref" href="topalg_5.html#T5" target="_self">Th5</a>, <a class="ref" href="topalg_5.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">Uit</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ui</font></span>)</span> <b>by </b><i><a class="ref" href="topalg_5.html#T4" target="_self">Th4</a>, , , <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E65:40_2_4_1_2_1"/><i><font color="Green">E468</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i>, <a class="ref" href="topalg_5.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E66:40_2_4_1_2_1"/><i><font color="Green">E469</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="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E67:40_2_4_1_2_1"/><i><font color="Green">E470</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E13:40_2"><i><font color="Green">E95</font></i></a>, , , <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E68:40_2_4_1_2_1"/><i><font color="Green">E471</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E13:40_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, , <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E69:40_2_4_1_2_1"/><i><font color="Green">E472</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4_1_2_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:40_2_4_1_2_1_7"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_2_4_1_2_1_7"/><i><font color="Green">E473</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:40_2_4_1_2_1_7"/><i><font color="Green">E474</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:40_2_4_1_2_1_7"/>
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E6:40_2_4_1_2_1_7"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fn1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:40_2_4_1_2_1_7"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="topalg_5.html#T4" target="_self">Th4</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E70:40_2_4_1_2_1"/><i><font color="Green">E475</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4_1_2_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:40_2_4_1_2_1_8"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_2_4_1_2_1_8"/><i><font color="Green">E476</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:40_2_4_1_2_1_8"/><i><font color="Green">E477</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:40_2_4_1_2_1_8"/>
<font color="Maroon">H</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E6:40_2_4_1_2_1_8"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:40_2_4_1_2_1_8"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="topalg_5.html#T4" target="_self">Th4</a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E71:40_2_4_1_2_1"/><i><font color="Green">E478</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"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T7" target="_self">Th7</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E72:40_2_4_1_2_1"/><i><font color="Green">E479</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"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E73:40_2_4_1_2_1"/>
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>  <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"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4_1_2_1_9"><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="E1:40_2_4_1_2_1_9"/><i><font color="Green">E480</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"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:40_2_4_1_2_1_9"/><i><font color="Green">E481</font></i>: 
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fn1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="topalg_5.html#T8" target="_self">Th8</a>, , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E3:40_2_4_1_2_1_9"/><i><font color="Green">E482</font></i>: 
<font color="Maroon">H</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, , <a class="ref" href="topalg_5.html#T8" target="_self">Th8</a>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="relat_1.html#T112">RELAT_1:112</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="topalg_5.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="topalg_5.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="topalg_5.html#T8" target="_self">Th8</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="topalg_5.html#T8" target="_self">Th8</a>, <a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</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/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="topalg_5.html#T10" target="_self">Th10</a>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, <a class="ref" href="topalg_5.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="topalg_5.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="relat_1.html#T112">RELAT_1:112</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E74:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E483</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_4_1_2_1_10"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_4_1_2_1_10"/>
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font> or  not <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_4_1_2_1_10_1"><b>suppose </b></a><a NAME="E1:40_2_4_1_2_1_10_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font>
 ;<br/><div class="add"><a NAME="E2:40_2_4_1_2_1_10_1"/><b>then </b><i><font color="Green">E484</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10:40_2"><i><font color="Green">E93</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="topalg_5.html#T8" target="_self">Th8</a>, <a class="ref" href="topalg_5.html#T5" target="_self">Th5</a>, <a class="ref" href="topalg_5.html#T7" target="_self">Th7</a>, , , , <a class="ref" href="funct_1.html#T49">FUNCT_1:49</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_4_1_2_1_10_2"><b>suppose </b></a><a NAME="E1:40_2_4_1_2_1_10_2"/>
not <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font>
 ;<br/><div class="add"><a NAME="E2:40_2_4_1_2_1_10_2"/><b>then </b>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9:40_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:40_2_4_1_2_1_10_2"/><b>then </b><i><font color="Green">E485</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E7:40_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_2"><i><font color="Green">E93</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="txt" href="topalg_5.html#E9:40"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E14:40_2_4_1_2"/>
<font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E39:40_2"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="txt" href="topalg_5.html#E4:40"><i><font color="Green">E78</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">Z</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E41:40_2"/><i><font color="Green">E486</font></i>: 
<font color="Maroon">Z</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"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">TT</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E42:40_2"/><i><font color="Green">E487</font></i>: 
<font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma57</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">Fn1</font> = 
<font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span>
<b>by </b><i>, , <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, , <a class="txt" href="topalg_5.html#E5:40"><i><font color="Green">E79</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>
<br/>.= 
<font color="Maroon">H</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, , <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E80</font></i></a>, , <a class="txt" href="topalg_5.html#E5:40"><i><font color="Green">E79</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>
;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E14:40"/><i><font color="Green">E488</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<a NAME="E15:40"/><i><font color="Green">E489</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="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E16:40"/><i><font color="Green">E490</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> holds  <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>, <font color="Maroon">t</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">N</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Maroon">Fn</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E2:40_3"/><i><font color="Green">E491</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">t</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E3:40_3"/><i><font color="Green">E492</font></i>: 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:40_3"/><i><font color="Green">E493</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E5:40_3"/>
<font color="Maroon">Fn</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E6:40_3"/><i><font color="Green">E494</font></i>: 
<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn</font>
 <b>and </b><br/><a NAME="E7:40_3"/>
<font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E8:40_3"/>
for <font color="Olive">H</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">H</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">H</font> &amp; <font color="Olive">H</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">Fn</font> <a href="funct_2.html#R2">=</a> <font color="Olive">H</font>
 <b>by </b><i/>;<br/>
<i><font color="Green">E495</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
;<br/>
<a NAME="E10:40_3"/><b>then </b><i><font color="Green">E496</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E1"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="topalg_5.html#E2"><i><font color="Green">Lemma56</font></i></a>, <a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E497</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">N</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> ;<br/><b>thus </b> <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E18:40"/><i><font color="Green">E498</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><br/> for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font><br/> for <font color="Olive">t</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">N1</font>, <font color="Olive">N2</font> being non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><br/> for <font color="Olive">Fn1</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Olive">N1</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <br/> for <font color="Olive">Fn2</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Olive">N2</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">t</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">N1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">N2</font> &amp; <font color="Olive">Fn2</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">Fn1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">Fn2</font> &amp; <font color="Olive">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">Fn1</font> &amp; <font color="Olive">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Olive">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">y</font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">y</font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/><b>let </b><font color="Maroon">N1</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Maroon">N2</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">Fn1</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">Fn2</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> ;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:40_5"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">t</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E2:40_5"/><i><font color="Green">E499</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 <b>and </b><br/><a NAME="E3:40_5"/><i><font color="Green">E500</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font>
 <b>and </b><br/><a NAME="E4:40_5"/><i><font color="Green">E501</font></i>: 
<font color="Maroon">Fn2</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E5:40_5"/><i><font color="Green">E502</font></i>: 
<font color="Maroon">Fn1</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E6:40_5"/><i><font color="Green">E503</font></i>: 
<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn2</font>
 <b>and </b><br/><a NAME="E7:40_5"/><i><font color="Green">E504</font></i>: 
<font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E8:40_5"/><i><font color="Green">E505</font></i>: 
<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font>
 <b>and </b><br/><a NAME="E9:40_5"/><i><font color="Green">E506</font></i>: 
<font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E10:40_5"/><i><font color="Green">E507</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">N1</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E11:40_5"/><i><font color="Green">E508</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fn1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:40_5"/><i><font color="Green">E509</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">N2</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E13:40_5"/><i><font color="Green">E510</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fn2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">TT</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E15:40_5"/><i><font color="Green">E511</font></i>: 
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E16:40_5"/><i><font color="Green">E512</font></i>: 
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">TT</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E17:40_5"/><i><font color="Green">E513</font></i>: 
<font color="Maroon">TT</font> is <a href="seqm_3.html#V1">increasing</a>
 <b>and </b><br/><a NAME="E18:40_5"/><i><font color="Green">E514</font></i>: 
 ex <font color="Olive">N</font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> &amp; ( for <font color="Olive">i</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font> &amp; <font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font> holds <br/> ex <font color="Olive">Ui</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="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Olive">Ui</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UL</font> &amp; <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">N</font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">Ui</font> ) ) )
 <b>by </b><i>, , , </i>;<br/>
<b>consider </b><font color="Maroon">N</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E20:40_5"/><i><font color="Green">E515</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E21:40_5"/><i><font color="Green">E516</font></i>: 
for <font color="Olive">i</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font> &amp; <font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font> holds <br/> ex <font color="Olive">Ui</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="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Olive">Ui</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UL</font> &amp; <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">Ui</font> )
 <b>by </b><i><a class="ref" href="topalg_5.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">N</font> = <font color="Maroon">N</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E23:40_5"/><i><font color="Green">E517</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E24:40_5"/><i><font color="Green">E518</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">TT</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E25:40_5"/><i><font color="Green">E519</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fn2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E26:40_5"/><i><font color="Green">E520</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E27:40_5"/><i><font color="Green">E521</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">N2</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E28:40_5"/><i><font color="Green">E522</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E29:40_5"/><i><font color="Green">E523</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">N1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E30:40_5"/><i><font color="Green">E524</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fn1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font> implies  ex <font color="Olive">Z</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Olive">Z</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"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Olive">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Olive">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> ) );<br/>
<a NAME="E31:40_5"/><i><font color="Green">E525</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_3.html#T26">FINSEQ_3:26</a></i>;<br/>
<a NAME="E32:40_5"/><i><font color="Green">E526</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1"><b>proof </b></a><div class="add">

<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><b>that </b><br/><a NAME="E1:40_5_1"/><i><font color="Green">E527</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>]
 <b>and </b><br/><a NAME="E2:40_5_1"/><i><font color="Green">E528</font></i>: 
<font color="Maroon">i</font> <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">TT</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_5_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font> )
 </span><b>by </b><i><a class="txt" href="topalg_5.html#E9:40"><i><font color="Green">E83</font></i></a>, <a class="ref" href="topreala.html#T23">TOPREALA:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1_1"><b>suppose </b></a><a NAME="E1:40_5_1_1_1"/><i><font color="Green">E529</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">Z</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>;<br/><a NAME="E2:40_5_1_1_1"/><i><font color="Green">E530</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a>, <a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>;<br/><b>reconsider </b><font color="Maroon">Z</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i>, , <a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a>, <a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>;<br/><b>take </b>
<font color="Maroon">Z</font>
;<br/><b>thus </b><a NAME="E4:40_5_1_1_1"/>
<font color="Maroon">Z</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"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><a NAME="E5:40_5_1_1_1"/><i><font color="Green">E531</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E7"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E6:40_5_1_1_1"/><b>then </b><i><font color="Green">E532</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E7:40_5_1_1_1"/><i><font color="Green">E533</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E8:40_5_1_1_1"/><b>then </b><i><font color="Green">E534</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E9:40_5_1_1_1"/>
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>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_1_1"><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="E1:40_5_1_1_1_1"/><i><font color="Green">E535</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"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:40_5_1_1_1_1"/><i><font color="Green">E536</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40"><i><font color="Green">E85</font></i></a>, , , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">Ft</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , , <a class="txt" href="topalg_5.html#E12:40"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">Fn2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40"><i><font color="Green">E85</font></i></a>, <a class="txt" href="topalg_5.html#E5"><i><font color="Green">Lemma59</font></i></a>, , , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E10:40_5_1_1_1"/>
<font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1_2"><b>suppose </b></a><a NAME="E1:40_5_1_1_2"/><i><font color="Green">E537</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">Z</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E3:40_5_1_1_2"/><i><font color="Green">E538</font></i>: 
<font color="Maroon">Z</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"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E4:40_5_1_1_2"/><i><font color="Green">E539</font></i>: 
<font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">ZZ</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>;<br/><div><i><font color="Green">E540</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_1"><b>now </b></a><div class="add"><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="E1:40_5_1_1_2_1"/><i><font color="Green">E541</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">TT</font>
 ;<br/><a NAME="E2:40_5_1_1_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:40_5_1_1_2_1"/><b>then </b>
( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> or 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:40_5_1_1_2_1"/><i><font color="Green">E542</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , <a class="txt" href="topalg_5.html#E1:40"><i><font color="Green">E72</font></i></a>, <a class="ref" href="seqm_3.html#D1">SEQM_3:def 1</a></i>;<br/><b>assume </b><a NAME="E5:40_5_1_1_2_1"/><i><font color="Green">E543</font></i>: 
<font color="Maroon">i</font> <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">TT</font>
 ;<br/><a NAME="E6:40_5_1_1_2_1"/><i><font color="Green">E544</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E7:40_5_1_1_2_1"/><i><font color="Green">E545</font></i>: 
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</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>, , , <a class="ref" href="seqm_3.html#D1">SEQM_3:def 1</a></i>;<br/><a NAME="E8:40_5_1_1_2_1"/>
<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">TT</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:40_5_1_1_2_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">TT</font> or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">TT</font> )
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E10:40_5_1_1_2_1"/>
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, , , <a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, <a class="ref" href="seqm_3.html#D1">SEQM_3:def 1</a></i>;<br/><b>hence </b><a NAME="E11:40_5_1_1_2_1"/>
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>thus </b><a NAME="E12:40_5_1_1_2_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</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>, , , , , <a class="ref" href="seqm_3.html#D1">SEQM_3:def 1</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E546</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_2"><b>now </b></a><div class="add"><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="E1:40_5_1_1_2_2"/><i><font color="Green">E547</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>thus </b><a NAME="E2:40_5_1_1_2_2"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_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="E1:40_5_1_1_2_2_1"/><i><font color="Green">E548</font></i>: 
<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"><span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:40_5_1_1_2_2_1"/>
( <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</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>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E4:40_5_1_1_2_2_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:40_5_1_1_2_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E7:40_5_1_1_2"/><i><font color="Green">E549</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a>, </i>;<br/><a NAME="E8:40_5_1_1_2"/><i><font color="Green">E550</font></i>: 
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="topalg_5.html#E9:40"><i><font color="Green">E83</font></i></a>, <a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a>, </i>;<br/><b>then reconsider </b><font color="Maroon">ZZ</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">Z1</font> = <font color="Maroon">Z</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">ZZ</font>;<br/><a NAME="E10:40_5_1_1_2"/><i><font color="Green">E551</font></i>: 
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</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><a class="txt" href="topalg_5.html#E9:40"><i><font color="Green">E83</font></i></a>, <a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a>, </i>;<br/><b>hence </b><a NAME="E11:40_5_1_1_2"/>
<font color="Maroon">Z1</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"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E11:40"><i><font color="Green">E85</font></i></a>, , <a class="ref" href="heine.html#T2">HEINE:2</a></i>;<br/><a NAME="E12:40_5_1_1_2"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E7"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E13:40_5_1_1_2"/><b>then </b><i><font color="Green">E552</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E14:40_5_1_1_2"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E15:40_5_1_1_2"/><b>then </b><i><font color="Green">E553</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, , <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E16:40_5_1_1_2"/>
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>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_3"><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="E1:40_5_1_1_2_3"/><i><font color="Green">E554</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"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_5_1_1_2_3"/><i><font color="Green">E555</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E4:40_5_1_1_2_3"/><i><font color="Green">E556</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font>
 <b>and </b><br/><a NAME="E5:40_5_1_1_2_3"/><i><font color="Green">E557</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xy</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:40_5_1_1_2_3"/><i><font color="Green">E558</font></i>: 
<font color="Maroon">xy</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N2</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E9"><i><font color="Green">Lemma64</font></i></a>, , <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E8:40_5_1_1_2_3"/><i><font color="Green">E559</font></i>: 
<font color="Maroon">xy</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N1</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E7"><i><font color="Green">Lemma62</font></i></a>, , <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E9:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E560</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E10:40_5_1_1_2_3"/><i><font color="Green">E561</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E11:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E562</font></i>: 
<font color="Maroon">xy</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xZZ</font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i>, , <a class="txt" href="topalg_5.html#E4:40"><i><font color="Green">E78</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>consider </b><font color="Maroon">Ui</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="toprealb.html#K8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E14:40_5_1_1_2_3"/><i><font color="Green">E563</font></i>: 
<font color="Maroon">Ui</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UL</font>
 <b>and </b><br/><a NAME="E15:40_5_1_1_2_3"/><i><font color="Green">E564</font></i>: 
<font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ui</font>
 <b>by </b><i>, <a class="txt" href="topalg_5.html#E9:40"><i><font color="Green">E83</font></i></a>, <a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a></i>;<br/>
<a NAME="E16:40_5_1_1_2_3"/><i><font color="Green">E565</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="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ui</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ui</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>consider </b><font color="Maroon">D</font> being  <a href="taxonom2.html#V4">mutually-disjoint</a> <a href="tops_2.html#V1">open</a> <a href="struct_0.html#NM3">Subset-Family</a> of <a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E18:40_5_1_1_2_3"/><i><font color="Green">E566</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">Ui</font>
 <b>and </b><br/><a NAME="E19:40_5_1_1_2_3"/><i><font color="Green">E567</font></i>: 
for <font color="Olive">d</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/>for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Olive">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ui</font></span>)</span>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Olive">d</font> holds <br/><font color="Olive">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, </i>;<br/>
<a NAME="E20:40_5_1_1_2_3"/><i><font color="Green">E568</font></i>: 
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40"><i><font color="Green">E85</font></i></a>, , <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E21:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E569</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:40_5_1_1_2_3"/><i><font color="Green">E570</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="txt" href="topalg_5.html#E9"><i><font color="Green">Lemma64</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E23:40_5_1_1_2_3"/><i><font color="Green">E571</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E24:40_5_1_1_2_3"/><i><font color="Green">E572</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N1</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E7"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>then </b><i><font color="Green">E573</font></i>: <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E26:40_5_1_1_2_3"/><i><font color="Green">E574</font></i>: 
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E27:40_5_1_1_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E28:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E575</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E29:40_5_1_1_2_3"/><b>then </b>
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">Ui</font>
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>then consider </b><font color="Maroon">Uit</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E31:40_5_1_1_2_3"/><i><font color="Green">E576</font></i>: 
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Uit</font>
 <b>and </b><br/><a NAME="E32:40_5_1_1_2_3"/><i><font color="Green">E577</font></i>: 
<font color="Maroon">Uit</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Uit</font> = <font color="Maroon">Uit</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E34:40_5_1_1_2_3"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="topalg_5.html#E9:40"><i><font color="Green">E83</font></i></a>, <a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E84</font></i></a>, </i>;<br/>
<a NAME="E35:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E578</font></i>: 
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E36:40_5_1_1_2_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">TT</font> <a href="seq_1.html#K2">.</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><a class="txt" href="topalg_5.html#E9:40"><i><font color="Green">E83</font></i></a>, </i>;<br/>
<a NAME="E37:40_5_1_1_2_3"/><b>then </b>
<font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E38:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E579</font></i>: 
<font color="Maroon">ZZ</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, , <a class="ref" href="borsuk_4.html#T49">BORSUK_4:49</a></i>;<br/>
<a NAME="E39:40_5_1_1_2_3"/>
<font color="Maroon">xy</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#T29">CONNSP_1:29</a></i>;<br/>
<a NAME="E40:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E580</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, <a class="ref" href="topalg_3.html#T17">TOPALG_3:17</a></i>;<br/>
<a NAME="E41:40_5_1_1_2_3"/><i><font color="Green">E581</font></i>: 
 <a href="topmetr.html#K5">I[01]</a>  is    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#T2">TSEP_1:2</a></i>;<br/>
<a NAME="E42:40_5_1_1_2_3"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T25">BORSUK_3:25</a></i>;<br/>
<a NAME="E43:40_5_1_1_2_3"/><b>then </b>
<font color="Maroon">xZZ</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>
<a NAME="E44:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E582</font></i>: 
<font color="Maroon">Fn1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="tops_2.html#T75">TOPS_2:75</a></i>;<br/>
<a NAME="E45:40_5_1_1_2_3"/><i><font color="Green">E583</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xy</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E46:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E584</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">xZZ</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E47:40_5_1_1_2_3"/>
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Fn1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E48:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E585</font></i>: 
<font color="Maroon">Uit</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Fn1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E49:40_5_1_1_2_3"/>
<font color="Maroon">D</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">Fn1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a>,<a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_5_1_1_2_3_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fn1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N1</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E3:40_5_1_1_2_3_2"/><i><font color="Green">E586</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xZZ</font>
 <b>and </b><br/><a NAME="E4:40_5_1_1_2_3_2"/><i><font color="Green">E587</font></i>: 
<font color="Maroon">Fn1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<i><font color="Green">E588</font></i>: <a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="txt" href="topalg_5.html#E4:40"><i><font color="Green">E78</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E6:40_5_1_1_2_3_2"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E7:40_5_1_1_2_3_2"/><b>then </b><i><font color="Green">E589</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:40_5_1_1_2_3_2"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E9:40_5_1_1_2_3_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E10:40_5_1_1_2_3_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="topalg_5.html#T5" target="_self">Th5</a>, , , , <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E50:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E590</font></i>: 
<font color="Maroon">Fn1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Uit</font>
 
<b>by </b><i>, <a class="ref" href="topalg_5.html#T3" target="_self">Th3</a>, <a class="ref" href="topalg_5.html#T4" target="_self">Th4</a>, <a class="ref" href="topalg_3.html#T13">TOPALG_3:13</a></i>;<br/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p3">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i>, <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, , , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E52:40_5_1_1_2_3"/><b>then </b>
<font color="Maroon">Fn2</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">Ui</font>
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>then consider </b><font color="Maroon">Uith</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E54:40_5_1_1_2_3"/><i><font color="Green">E591</font></i>: 
<font color="Maroon">Fn2</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Uith</font>
 <b>and </b><br/><a NAME="E55:40_5_1_1_2_3"/><i><font color="Green">E592</font></i>: 
<font color="Maroon">Uith</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Uith</font> = <font color="Maroon">Uith</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topalg_5.html#T5" target="_self">Th5</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">XZZ</font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i>, <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, , <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E58:40_5_1_1_2_3"/>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="borsuk_3.html#T25">BORSUK_3:25</a></i>;<br/>
<a NAME="E59:40_5_1_1_2_3"/><b>then </b>
<font color="Maroon">XZZ</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>
<a NAME="E60:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E594</font></i>: 
<font color="Maroon">Fn2</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">XZZ</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="tops_2.html#T75">TOPS_2:75</a></i>;<br/>
<a NAME="E61:40_5_1_1_2_3"/><i><font color="Green">E595</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E62:40_5_1_1_2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E63:40_5_1_1_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fn2</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="topalg_5.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E64:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E596</font></i>: 
<font color="Maroon">Fn2</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Fn2</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E65:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E597</font></i>: 
<font color="Maroon">Uith</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Fn2</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E66:40_5_1_1_2_3"/>
<font color="Maroon">D</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">Fn2</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a>,<a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_5_1_1_2_3_4"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fn2</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N2</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E3:40_5_1_1_2_3_4"/><i><font color="Green">E598</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xZZ</font>
 <b>and </b><br/><a NAME="E4:40_5_1_1_2_3_4"/><i><font color="Green">E599</font></i>: 
<font color="Maroon">Fn2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<i><font color="Green">E600</font></i>: <a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a>, <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E6:40_5_1_1_2_3_4"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E7:40_5_1_1_2_3_4"/><b>then </b><i><font color="Green">E601</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:40_5_1_1_2_3_4"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E9:40_5_1_1_2_3_4"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="topalg_5.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E10:40_5_1_1_2_3_4"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="topalg_5.html#T8" target="_self">Th8</a>, , , , <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E67:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E602</font></i>: 
<font color="Maroon">Fn2</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Uith</font>
 
<b>by </b><i>, , , <a class="ref" href="topalg_3.html#T13">TOPALG_3:13</a></i>;<br/>
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">Fn2</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="txt" href="topalg_5.html#E12:40"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E69:40_5_1_1_2_3"/><b>then </b>
<font color="Maroon">Uit</font> <a href="subset_1.html#R2">meets</a> <font color="Maroon">Uith</font>
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E70:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E603</font></i>: 
<font color="Maroon">Uit</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Uith</font>
 
<b>by </b><i>, , <a class="ref" href="taxonom2.html#D5">TAXONOM2:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">f</font> = <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font>;<br/>
<a NAME="E71:40_5_1_1_2_3"/><i><font color="Green">E604</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">Uit</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Uit</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E72:40_5_1_1_2_3"/><i><font color="Green">E605</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Uit</font>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E73:40_5_1_1_2_3"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ui</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_3_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:40_5_1_1_2_3_6"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_5_1_1_2_3_6"/><i><font color="Green">E606</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font></span>)</span>
 <b>and </b><br/><a NAME="E4:40_5_1_1_2_3_6"/><i><font color="Green">E607</font></i>: 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:40_5_1_1_2_3_6"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="topalg_5.html#T10" target="_self">Th10</a>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:40_5_1_1_2_3_6"/><b>then </b>
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ui</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<b>hence </b><a NAME="E7:40_5_1_1_2_3_6"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ui</font>
 <b>by </b><i>, , <a class="ref" href="topalg_5.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">Uit</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">Uit</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Ui</font></span>)</span> <b>by </b><i><a class="ref" href="topalg_5.html#T10" target="_self">Th10</a>, , , <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E75:40_5_1_1_2_3"/><i><font color="Green">E608</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="ref" href="topalg_5.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E76:40_5_1_1_2_3"/><i><font color="Green">E609</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="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E77:40_5_1_1_2_3"/><i><font color="Green">E610</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="txt" href="topalg_5.html#E5:40"><i><font color="Green">E79</font></i></a>, , <a class="txt" href="topalg_5.html#E4:40"><i><font color="Green">E78</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E78:40_5_1_1_2_3"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N2</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E79:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E611</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, , <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E80:40_5_1_1_2_3"/><i><font color="Green">E612</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_3_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:40_5_1_1_2_3_7"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_5_1_1_2_3_7"/><i><font color="Green">E613</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:40_5_1_1_2_3_7"/><i><font color="Green">E614</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:40_5_1_1_2_3_7"/>
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T11" target="_self">Th11</a>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E6:40_5_1_1_2_3_7"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fn1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E5:40"><i><font color="Green">E79</font></i></a>, <a class="ref" href="topalg_5.html#T11" target="_self">Th11</a>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:40_5_1_1_2_3_7"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="topalg_5.html#T10" target="_self">Th10</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E81:40_5_1_1_2_3"/><i><font color="Green">E615</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_3_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:40_5_1_1_2_3_8"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_5_1_1_2_3_8"/><i><font color="Green">E616</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:40_5_1_1_2_3_8"/><i><font color="Green">E617</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:40_5_1_1_2_3_8"/>
<font color="Maroon">Fn2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T12" target="_self">Th12</a>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E6:40_5_1_1_2_3_8"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fn2</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">xZZ</font>
 
<b>by </b><i>, <a class="ref" href="topalg_5.html#T12" target="_self">Th12</a>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:40_5_1_1_2_3_8"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="topalg_5.html#T10" target="_self">Th10</a>, , <a class="ref" href="topalg_5.html#T8" target="_self">Th8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E82:40_5_1_1_2_3"/><i><font color="Green">E618</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"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E83:40_5_1_1_2_3"/><i><font color="Green">E619</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"><font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E84:40_5_1_1_2_3"/>
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>  <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"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_3_9"><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="E1:40_5_1_1_2_3_9"/><i><font color="Green">E620</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"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:40_5_1_1_2_3_9"/><i><font color="Green">E621</font></i>: 
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fn1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="topalg_5.html#T13" target="_self">Th13</a>, <a class="ref" href="topalg_5.html#T11" target="_self">Th11</a>, <a class="txt" href="topalg_5.html#E5:40"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E3:40_5_1_1_2_3_9"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">Fn1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E4:40_5_1_1_2_3_9"/><b>then </b><i><font color="Green">E622</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T13" target="_self">Th13</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:40_5_1_1_2_3_9"/><i><font color="Green">E623</font></i>: 
<font color="Maroon">Fn2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fn2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="topalg_5.html#T13" target="_self">Th13</a>, <a class="ref" href="topalg_5.html#T11" target="_self">Th11</a>, <a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="topalg_5.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:40_5_1_1_2_3_9"/><i><font color="Green">E624</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fn1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="relat_1.html#T112">RELAT_1:112</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="topalg_5.html#T11" target="_self">Th11</a>, , <a class="ref" href="topalg_5.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E5:40"><i><font color="Green">E79</font></i></a>, <a class="ref" href="topalg_5.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E24"><i><font color="Green">Lemma146</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="topalg_5.html#T13" target="_self">Th13</a>, <a class="txt" href="topalg_5.html#E5:40"><i><font color="Green">E79</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">x</font>
<b>by </b><i><a class="ref" href="topalg_5.html#T17" target="_self">Th17</a>, <a class="ref" href="topalg_5.html#T13" target="_self">Th13</a>, , <a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="topalg_5.html#T14" target="_self">Th14</a>, <a class="ref" href="topalg_5.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="topalg_5.html#T14" target="_self">Th14</a>, <a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="topalg_5.html#T7" target="_self">Th7</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> <span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="topalg_5.html#T16" target="_self">Th16</a>, , <a class="ref" href="topalg_5.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="topalg_5.html#T14" target="_self">Th14</a>, <a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="topalg_5.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn2</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="topalg_5.html#T11" target="_self">Th11</a>, , <a class="ref" href="topalg_5.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="relat_1.html#T112">RELAT_1:112</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E85:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E625</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T11" target="_self">Th11</a>, <a class="ref" href="topalg_5.html#T12" target="_self">Th12</a>, , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1_2_3_10"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_5_1_1_2_3_10"/>
( <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font> or  not <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1_2_3_10_1"><b>suppose </b></a><a NAME="E1:40_5_1_1_2_3_10_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font>
 ;<br/><div class="add"><a NAME="E2:40_5_1_1_2_3_10_1"/><b>then </b><i><font color="Green">E626</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E24"><i><font color="Green">Lemma146</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">xy</font>,<font color="Maroon">ZZ</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="topalg_5.html#T13" target="_self">Th13</a>, , , , <a class="ref" href="topalg_5.html#T11" target="_self">Th11</a>, <a class="ref" href="topalg_5.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#T49">FUNCT_1:49</a></i>
<br/>.= 
<font color="Maroon">Fn2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E24"><i><font color="Green">Lemma146</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1_2_3_10_2"><b>suppose </b></a><a NAME="E1:40_5_1_1_2_3_10_2"/>
not <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font>
 ;<br/><div class="add"><a NAME="E2:40_5_1_1_2_3_10_2"/><b>then </b>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:40_5_1_1_2_3_10_2"/><b>then </b><i><font color="Green">E627</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">Fn1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E24"><i><font color="Green">Lemma146</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">Fn2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E24"><i><font color="Green">Lemma146</font></i></a>, <a class="txt" href="topalg_5.html#E12:40"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E17:40_5_1_1_2"/>
<font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E33:40_5"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E80</font></i></a>, <a class="txt" href="topalg_5.html#E8:40"><i><font color="Green">E81</font></i></a>);<br/></i>
<a NAME="E34:40_5"/><b>then </b>
 ex <font color="Olive">Z</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Olive">Z</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"><font color="Maroon">TT</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">TT</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Olive">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Olive">Z</font></span><a href="borsuk_1.html#K6">:]</a></span> )
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a></i>;<br/>
<b>hence </b><a NAME="E35:40_5"/>
<font color="Maroon">Fn1</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn2</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:40"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><br/> for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>   st <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> holds <br/> ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">G</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:40_6"/><i><font color="Green">E628</font></i>: 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E2:40_6"/><i><font color="Green">E629</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>, <font color="Maroon">t</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">N</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Maroon">Fn</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E4:40_6"/><i><font color="Green">E630</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">t</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E5:40_6"/><i><font color="Green">E631</font></i>: 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E6:40_6"/><i><font color="Green">E632</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E7:40_6"/><i><font color="Green">E633</font></i>: 
<font color="Maroon">Fn</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E8:40_6"/><i><font color="Green">E634</font></i>: 
<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn</font>
 <b>and </b><br/><a NAME="E9:40_6"/><i><font color="Green">E635</font></i>: 
<font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E10:40_6"/>
for <font color="Olive">H</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">H</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">H</font> &amp; <font color="Olive">H</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">Fn</font> <a href="funct_2.html#R2">=</a> <font color="Olive">H</font>
 <b>by </b><i/>;<br/>
<i><font color="Green">E636</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
;<br/>
<a NAME="E12:40_6"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2"><i><font color="Green">Lemma56</font></i></a>, <a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E14:40_6"/><i><font color="Green">E637</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E15:40_6"/><i><font color="Green">E638</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E16:40_6"/><i><font color="Green">E639</font></i>: 
<font color="Maroon">Fn</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="txt" href="topalg_5.html#E1"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="topalg_5.html#E5"><i><font color="Green">Lemma59</font></i></a>, <a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<a NAME="E17:40_6"/><i><font color="Green">E640</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E7"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:40_6"/>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="borsuk_3.html#T26">BORSUK_3:26</a></i>;<br/>
<b>then consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E20:40_6"/><i><font color="Green">E641</font></i>: 
<font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E21:40_6"/><i><font color="Green">E642</font></i>: 
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>take </b><font color="Maroon">WW</font> = <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>;<br/>

<a NAME="E22:40_6"/><i><font color="Green">E644</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a> <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="E23:40_6"/><b>then </b><i><font color="Green">E645</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T1">BORSUK_3:1</a></i>;<br/>
<b>thus </b><a NAME="E24:40_6"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">WW</font>
 <b>by </b><i>, <a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, <a class="ref" href="topalg_5.html#T1" target="_self">Th1</a>, <a class="ref" href="borsuk_3.html#T1">BORSUK_3:1</a></i>;<br/>

<a NAME="E25:40_6"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/>
<b>hence </b><a NAME="E26:40_6"/>
<font color="Maroon">WW</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>

<b>let </b><font color="Maroon">y</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="E27:40_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">WW</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E29:40_6"/><i><font color="Green">E646</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">WW</font>
 <b>and </b><br/><a NAME="E30:40_6"/><i><font color="Green">E647</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon">y0</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>, <font color="Maroon">t0</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">N0</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Maroon">Fn0</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N0</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E32:40_6"/><i><font color="Green">E652</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y0</font>,<font color="Maroon">t0</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E33:40_6"/><i><font color="Green">E653</font></i>: 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn0</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E34:40_6"/><i><font color="Green">E654</font></i>: 
<font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N0</font>
 <b>and </b><br/><a NAME="E35:40_6"/><i><font color="Green">E655</font></i>: 
<font color="Maroon">Fn0</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E36:40_6"/><i><font color="Green">E656</font></i>: 
<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N0</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn0</font>
 <b>and </b><br/><a NAME="E37:40_6"/><i><font color="Green">E657</font></i>: 
<font color="Maroon">Fn0</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N0</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E38:40_6"/>
for <font color="Olive">H</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N0</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">H</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N0</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">H</font> &amp; <font color="Olive">H</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N0</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">Fn0</font> <a href="funct_2.html#R2">=</a> <font color="Olive">H</font>
 <b>by </b><i/>;<br/>
<a NAME="E39:40_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">N</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E40:40_6"/><b>then </b><i><font color="Green">E658</font></i>: 
<font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E41:40_6"/><i><font color="Green">E659</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y0</font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, <a class="ref" href="zfmisc_1.html#T128">ZFMISC_1:128</a></i>;<br/>
<b>then </b><font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y0</font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Fn0</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y0</font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E5:40"><i><font color="Green">E79</font></i></a>, <a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E5"><i><font color="Green">Lemma59</font></i></a>, <a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E75</font></i></a>, , , , , <a class="txt" href="topalg_5.html#E4:40"><i><font color="Green">E78</font></i></a>, </i>
<br/>.= 
<font color="Maroon">Fn0</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E80</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E43:40_6"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fn</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40"><i><font color="Green">E72</font></i></a>, , , , , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E44:40_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:40"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>

<b>thus </b><a NAME="E21:40"/>
<font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>

<a NAME="E22:40"/><i><font color="Green">E660</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E23:40"/><i><font color="Green">E661</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">Ft</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, , , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E24:40"/>
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>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Ft</font> holds <br/><font color="Maroon">Ft</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_7"><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="E1:40_7"/><i><font color="Green">E662</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">Ft</font>
 ;<br/>

<a NAME="E2:40_7"/><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">G</font>
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>, <font color="Maroon">t</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">N</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Maroon">Fn</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E4:40_7"/><i><font color="Green">E663</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">t</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E5:40_7"/><i><font color="Green">E664</font></i>: 
<font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E6:40_7"/><i><font color="Green">E665</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E7:40_7"/>
( <font color="Maroon">Fn</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn</font> )
 <b>and </b><br/><a NAME="E8:40_7"/><i><font color="Green">E666</font></i>: 
<font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E9:40_7"/>
for <font color="Olive">H</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">H</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">H</font> &amp; <font color="Olive">H</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">Fn</font> <a href="funct_2.html#R2">=</a> <font color="Olive">H</font>
 <b>by </b><i/>;<br/>
<a NAME="E10:40_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma57</font></i></a>, <a class="txt" href="topalg_5.html#E5"><i><font color="Green">Lemma59</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><font color="Maroon">Ft</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="txt" href="topalg_5.html#E2"><i><font color="Green">Lemma56</font></i></a>, , <a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E25:40"/>
<font color="Maroon">G</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>

<b>let </b><font color="Maroon">H</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E26:40"/><i><font color="Green">E667</font></i>: 
<font color="Maroon">H</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E27:40"/><i><font color="Green">E668</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E28:40"/><i><font color="Green">E669</font></i>: 
<font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font>
 ;<br/>

<a NAME="E29:40"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> holds  <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>, <font color="Maroon">t</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">N</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Maroon">Fn</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E2:40_8"/><i><font color="Green">E670</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">t</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E3:40_8"/><i><font color="Green">E671</font></i>: 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:40_8"/><i><font color="Green">E672</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E5:40_8"/>
( <font color="Maroon">Fn</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">Fn</font> )
 <b>and </b><br/><a NAME="E6:40_8"/>
<font color="Maroon">Fn</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E7:40_8"/><i><font color="Green">E673</font></i>: 
for <font color="Olive">H</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>   st <font color="Olive">H</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">H</font> &amp; <font color="Olive">H</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">Fn</font> <a href="funct_2.html#R2">=</a> <font color="Olive">H</font>
 <b>by </b><i/>;<br/>
<i><font color="Green">E674</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
;<br/>
<a NAME="E9:40_8"/><b>then </b><i><font color="Green">E675</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E5"><i><font color="Green">Lemma59</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E10:40_8"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:40_8"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E12:40_8"/><b>then </b><i><font color="Green">E676</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E13:40_8"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3">R^1</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">H1</font> = <font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">N</font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E10"><i><font color="Green">Lemma65</font></i></a>, <a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E15:40_8"/><i><font color="Green">E678</font></i>: 
<font color="Maroon">H1</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="topalg_3.html#T18">TOPALG_3:18</a></i>;<br/>
<i><font color="Green">E679</font></i>: <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> = 
<font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>
<br/>.= 
<font color="Maroon">Ft</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E4"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>
;<br/>
<a NAME="E17:40_8"/>
<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="relat_1.html#T112">RELAT_1:112</a></i>;<br/>
<b>hence </b><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">N</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E7"><i><font color="Green">Lemma62</font></i></a>, , , </i>
<br/>.= 
<font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma63</font></i></a>, <a class="txt" href="topalg_5.html#E9"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E30:40"/>
<font color="Maroon">G</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
