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

<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"><a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:90"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="real_1.html#NM1">Real</a> holds  <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="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="finseq_4.html#K2">*&gt;</a></span>
 ;<br/>

<a NAME="E2:90"/><i><font color="Green">E57</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"><a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</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 <a href="topmetr.html#K3">R^1</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</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="E3:90"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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 <a href="topmetr.html#K3">R^1</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</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/>
<b>hence </b> <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> = 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K3">R^1</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K3">R^1</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>

<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> </span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_3.html#T1">BORSUK_3:1</a></i>
;<br/> <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a><br/>

<b>thus </b><a NAME="E5:90"/><i><font color="Green">E59</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"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:90_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ; <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">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="E2:90_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:90_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T55">EUCLID:55</a></i>;<br/>
<a NAME="E5:90_2"/><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></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">f</font>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:90_2"/>
<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">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:90_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


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

<b>thus </b><a NAME="E6:90"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <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="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:90_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then 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="E3:90_3"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  )
 <b>and </b><br/><a NAME="E4:90_3"/><i><font color="Green">E71</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="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>assume </b><a NAME="E5:90_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then 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="E7:90_3"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  )
 <b>and </b><br/><a NAME="E8:90_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">y</font> <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="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font>, <font color="Maroon">x2</font> = <font color="Maroon">x2</font>, <font color="Maroon">y1</font> = <font color="Maroon">y1</font>, <font color="Maroon">y2</font> = <font color="Maroon">y2</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, </i>;<br/>
<a NAME="E10:90_3"/><i><font color="Green">E84</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>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> &amp; <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">y1</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> )
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E11:90_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E12:90_3"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> )
 
<b>by </b><i>, <a class="txt" href="topreal6.html#E1"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="topreal6.html#T6" target="_self">Th6</a>, <a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>;<br/>
<b>hence </b><a NAME="E13:90_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="txt" href="topreal6.html#E1"><i><font color="Green">Lemma54</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E7:90"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> </span><a href="borsuk_1.html#K5">:]</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D3">BORSUK_1:def 3</a><br/><b>let </b><font color="Maroon">G</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>;<br/>



<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:90_4"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  )
 <b>and </b><br/><a NAME="E3:90_4"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</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">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fw</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E6:90_4"/>
<font color="Maroon">fw</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E8:90_4"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E9:90_4"/><i><font color="Green">E88</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">fw</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="goboard6.html#T8">GOBOARD6:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>, <font color="Maroon">B</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>take </b>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span>
;<br/>

<b>thus </b><a NAME="E12:90_4"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">w</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:90_4_1"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="jordan6.html#T46">JORDAN6:46</a></i>;<br/>
<a NAME="E2:90_4_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a>  <a href="borsuk_1.html#K10">Base-Appr</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
;<br/>
<a NAME="E3:90_4_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E4:90_4_1"/>
<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">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:90_4_1"/><b>then </b>
<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">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="euclid.html#K23">]|</a></span>
 
;<br/>
<a NAME="E6:90_4_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E7:90_4_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="txt" href="topreal6.html#E1:90"><i><font color="Green">E56</font></i></a>, </i>;<br/>
<a NAME="E8:90_4_1"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</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="E9:90_4_1"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K10">Base-Appr</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal6.html#T6" target="_self">Th6</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E10:90_4_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T55">BORSUK_1:55</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a><br/>


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

<a NAME="E13:90_4"/>
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">fw</font>,<font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:90_4"/><b>then </b>
<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">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">fw</font>,<font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E15:90_4"/>
<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">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="topreal6.html#E1"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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

<a NAME="E8:90"/><i><font color="Green">E91</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> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="tops_2.html#T62">TOPS_2:62</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> =  <a href="pscomp_1.html#K13">proj1</a> , <font color="Maroon">f2</font> =  <a href="pscomp_1.html#K14">proj2</a>  as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E10:90"/><i><font color="Green">E92</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="funct_3.html#K14">&lt;:</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="funct_3.html#K14">:&gt;</a></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="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_5"><b>now </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="E1:90_5"/><i><font color="Green">E93</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="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:90_5"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T55">EUCLID:55</a></i>;<br/><a NAME="E4:90_5"/><i><font color="Green">E95</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:90_5"/><i><font color="Green">E96</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></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">f</font>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:90_5"/><i><font color="Green">E97</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">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/><a NAME="E8:90_5"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="topreal6.html#E1"><i><font color="Green">Lemma54</font></i></a></i>;<br/><b>thus </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="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="txt" href="topreal6.html#E1"><i><font color="Green">Lemma54</font></i></a>, <a class="txt" href="topreal6.html#E1:90"><i><font color="Green">E56</font></i></a>, <a class="txt" href="topreal6.html#E2:90"><i><font color="Green">E57</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="topreal6.html#E3:90"><i><font color="Green">E58</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="topreal6.html#E3:90"><i><font color="Green">E58</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>
<br/>.= 
<span class="p1"><a href="funct_3.html#K14">&lt;:</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="funct_3.html#K14">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="topreal6.html#T6" target="_self">Th6</a>, <a class="ref" href="funct_3.html#T69">FUNCT_3:69</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E12:90"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="funct_2.html#R1">=</a> <span class="p1"><a href="funct_3.html#K14">&lt;:</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="funct_3.html#K14">:&gt;</a></span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E13:90"/>
( <font color="Maroon">f1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">f2</font> is <a href="pre_topc.html#V5">continuous</a> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E14:90"/>
<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="yellow12.html#T41">YELLOW12:41</a></i>;<br/>


</div>
