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

<b>set </b><font color="Maroon">D1</font> =  <a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1;<br/>
<b>set </b><font color="Maroon">D2</font> =  <a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span>;<br/>
<a NAME="E1:117"/><i><font color="Green">E61</font></i>: 
 <a href="toprealb.html#K9" target="_self">c[10]</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="toprealb.html#K9" target="_self">c[10]</a> </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/>
<b>set </b><font color="Maroon">F1</font> = <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">F2</font> = <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">F</font> = <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p4">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p4">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span>;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p4">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p4">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p5">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7">}</a></span> as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<a NAME="E3:117"/><i><font color="Green">E66</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">0 <a href="real_1.html#K3">+</a> 0</span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> 0</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R2">in</a>  <a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1
 
<b>by </b><i/>;<br/>
<a NAME="E4:117"/><b>then </b><i><font color="Green">E68</font></i>: 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:117"/>
<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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R2">in</a>  <a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:117"/><b>then </b><i><font color="Green">E69</font></i>: 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:117"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="toprealb.html#T12" target="_self">Th12</a>, </i>;<br/>

<b>thus </b><a NAME="E8:117"/>
<font color="Maroon">F</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:117_1"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>,<a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_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="E1:117_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">A</font> = <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">0 <a href="real_1.html#K3">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1">.[</a></span> as   <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="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">A</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">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E8"><i><font color="Green">Lemma51</font></i></a></i>;<br/>
<a NAME="E4:117_1_1"/>
<font color="Maroon">f</font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E10"><i><font color="Green">Lemma53</font></i></a></i>;<br/>
<a NAME="E5:117_1_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</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="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:117_1_1"/><i><font color="Green">E72</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">f</font>
 <b>and </b><br/><a NAME="E8:117_1_1"/><i><font color="Green">E73</font></i>: 
<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">a</font>
 <b>by </b><i><a class="ref" href="toprealb.html#T14" target="_self">Th14</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:117_1_1"/><i><font color="Green">E74</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E6"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="toprealb.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E11:117_1_1"/><i><font color="Green">E75</font></i>: 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <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">x</font>
 
<b>by </b><i><a class="ref" href="toprealb.html#T15" target="_self">Th15</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E12:117_1_1"/><i><font color="Green">E77</font></i>: 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:117_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) or ( <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) )
 </span><b>by </b><i><a class="ref" href="toprealb.html#T15" target="_self">Th15</a>, <a class="ref" href="toprealb.html#T19" target="_self">Th19</a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1_1_1"><b>suppose </b></a><a NAME="E1:117_1_1_1_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:117_1_1_1_1"/>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><b>then </b><i><font color="Green">E116</font></i>: <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 = 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="toprealb.html#K9" target="_self">c[10]</a> 
<b>by </b><i/>
<br/>.= 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> 1
<b>by </b><i/>
;<br/><a NAME="E4:117_1_1_1_1"/>
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E5:117_1_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, , <a class="txt" href="toprealb.html#E1:117_1_1_1_1"><i><font color="Green">E115</font></i></a>, <a class="txt" href="toprealb.html#E6"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>hence </b><a NAME="E6:117_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="toprealb.html#T13" target="_self">Th13</a>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1_1_2"><b>suppose </b></a><a NAME="E1:117_1_1_1_2"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:117_1_1_1_2"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E3:117_1_1_1_2"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">0 <a href="real_1.html#K3">+</a> 0</span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> 0</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R2">in</a>  <a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1
 <b>by </b><i/>;<br/><a NAME="E4:117_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1</span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E5:117_1_1_1_2"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="txt" href="toprealb.html#E6"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:117_1_1_1_2"/>
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:117_1_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1_1_3"><b>suppose </b></a><a NAME="E1:117_1_1_1_3"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:117_1_1_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E2:117_1"/>
 <a href="setfam_1.html#K5">union</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="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span>
 ;<br/>


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

<b>thus </b><a NAME="E9:117"/>
<font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<a NAME="E1:117_2"/><i><font color="Green">E121</font></i>: 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K13" target="_self">CircleMap</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4" target="_self">R^1</a> 0</span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="extens_1.html#T1">EXTENS_1:1</a></i>;<br/>
<a NAME="E2:117_2"/><i><font color="Green">E124</font></i>: 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K13" target="_self">CircleMap</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4" target="_self">R^1</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="extens_1.html#T1">EXTENS_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = 0 as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<div><i><font color="Green">E127</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><a NAME="E1:117_2_1"/><i><font color="Green">E128</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> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E2:117_2_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/><b>hence </b><a NAME="E3:117_2_1"/>
<font color="Maroon">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="toprealb.html#T19" target="_self">Th19</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">M</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">r</font>,<span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">N</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<div><i><font color="Green">E129</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">A</font></span>)</span></span>)</span> <b>by </b><i/>;<br/><a NAME="E2:117_2_2"/><i><font color="Green">E130</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> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E3:117_2_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:117_2_2"/>
<font color="Maroon">A</font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:117_2"/><b>then </b>
<font color="Maroon">M</font> is <a href="pre_topc.html#V3">open</a>
 
;<br/>
<a NAME="E9:117_2"/><b>then </b><i><font color="Green">E131</font></i>: 
<span class="p1">(<span class="default"><a href="toprealb.html#K13" target="_self">CircleMap</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4" target="_self">R^1</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">M</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="t_0topsp.html#D2">T_0TOPSP:def 2</a></i>;<br/>
<a NAME="E10:117_2"/>
<font color="Maroon">N</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="toprealb.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E11:117_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="toprealb.html#K13" target="_self">CircleMap</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4" target="_self">R^1</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="t_0topsp.html#D2">T_0TOPSP:def 2</a></i>;<br/>
<a NAME="E12:117_2"/><b>then </b>
( <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1</span>)</span></span>)</span> is <a href="pre_topc.html#V3">open</a> &amp; <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, <a class="ref" href="toprealb.html#T13" target="_self">Th13</a>, <a class="ref" href="toprealb.html#T14" target="_self">Th14</a>, <a class="ref" href="toprealb.html#T15" target="_self">Th15</a>, , <a class="ref" href="tsep_1.html#T17">TSEP_1:17</a></i>;<br/>
<b>hence </b><a NAME="E13:117_2"/>
(  not <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> or <font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


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

<b>let </b><font color="Maroon">U</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span>;<br/>

<b>thus </b><a NAME="E10:117"/>
( <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span> implies (  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1</span>)</span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">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>  <a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1 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" target="_self">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U</font></span>)</span>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">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>  ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:117_3"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<a NAME="E2:117_3"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">0 <a href="real_1.html#K3">+</a> 0</span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> 0</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R2">in</a>  <a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1
 
<b>by </b><i/>;<br/>
<a NAME="E3:117_3"/><b>then </b><i><font color="Green">E133</font></i>: 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="topalg_2.html#K3">R^1</a> , <font color="Maroon">x2</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/><b>assume </b><a NAME="E1:117_3_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1</span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:117_3_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E4:117_3_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E6:117_3_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="toprealb.html#T19" target="_self">Th19</a></i>;<br/><b>assume </b><a NAME="E7:117_3_1"/><i><font color="Green">E179</font></i>: 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font>
 ;<br/><b>set </b><font color="Maroon">k</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>;<br/><b>set </b><font color="Maroon">K</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K2">+</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/><a NAME="E8:117_3_1"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><a NAME="E9:117_3_1"/><b>then </b><i><font color="Green">E180</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K2">+</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R2">in</a>  <a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1
 ;<br/><a NAME="E10:117_3_1"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font>
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E11:117_3_1"/>
<font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E12:117_3_1"/><b>then </b><i><font color="Green">E182</font></i>: 
<font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E13:117_3_1"/><b>then </b><i><font color="Green">E191</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x1</font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><a NAME="E14:117_3_1"/><i><font color="Green">E192</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><a NAME="E15:117_3_1"/>
not <font color="Maroon">x1</font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i>, , <a class="ref" href="int_1.html#T48">INT_1:48</a></i>;<br/><a NAME="E16:117_3_1"/><b>then </b><i><font color="Green">E193</font></i>: 
not <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_3_1_1"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/><a NAME="E2:117_3_1_1"/><b>then </b>
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="toprealb.html#K9" target="_self">c[10]</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:117_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="toprealb.html#E34"><i><font color="Green">Lemma104</font></i></a>, , <a class="txt" href="toprealb.html#E6"><i><font color="Green">Lemma43</font></i></a>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/></div><b>end;</b></div><a NAME="E18:117_3_1"/><b>then </b><i><font color="Green">E194</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x2</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E19:117_3_1"/><i><font color="Green">E195</font></i>: 
( <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E20:117_3_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/><a NAME="E21:117_3_1"/><b>then </b><i><font color="Green">E196</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E22:117_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E23:117_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E24:117_3_1"/><b>then </b><i><font color="Green">E197</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E25:117_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/><a NAME="E26:117_3_1"/><b>then </b><i><font color="Green">E198</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E27:117_3_1"/>
<font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><a NAME="E28:117_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E29:117_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E30:117_3_1"/><b>then </b><i><font color="Green">E260</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E31:117_3_1"/>
( <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K20">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p5"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K20">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p5"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="txt" href="toprealb.html#E5"><i><font color="Green">Lemma42</font></i></a></i>;<br/><a NAME="E32:117_3_1"/><b>then </b>
(  <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> &amp;  <a href="sin_cos.html#K20">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K20">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="sppol_2.html#T1">SPPOL_2:1</a></i>;<br/><a NAME="E33:117_3_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="complex2.html#T12">COMPLEX2:12</a></i>;<br/><a NAME="E34:117_3_1"/><b>then </b>
<font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E35:117_3_1"/><b>then </b><i><font color="Green">E261</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 ;<br/><a NAME="E36:117_3_1"/>
<font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E37:117_3_1"/><b>then </b>
<font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E38:117_3_1"/><b>then </b>
<font color="Maroon">x2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E39:117_3_1"/><b>then </b>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K2">+</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E40:117_3_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1</span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:117_3"/>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1</span>)</span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">U</font>
 <b>by </b><i><a class="ref" href="toprealb.html#T14" target="_self">Th14</a>, <a class="ref" href="toprealb.html#T15" target="_self">Th15</a>, <a class="ref" href="t_0topsp.html#T2">T_0TOPSP:2</a></i>;<br/>

<b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/>

<b>assume </b><a NAME="E6:117_3"/><i><font color="Green">E262</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <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">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U</font></span>)</span>;<br/>

<b>assume </b><a NAME="E7:117_3"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">d</font>
 ;<br/>

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E9:117_3"/><i><font color="Green">E264</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E10:117_3"/>
( 0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">d</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="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E12:117_3"/><i><font color="Green">E265</font></i>: 
1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">0 <a href="real_1.html#K3">+</a> 0</span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K3">+</a> 0</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<b>reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">U</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="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> <b>by </b><i><a class="ref" href="toprealb.html#T14" target="_self">Th14</a>, , <a class="txt" href="toprealb.html#E6"><i><font color="Green">Lemma43</font></i></a>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</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">d1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U1</font></span>)</span> ;<br/>
<a NAME="E15:117_3"/><i><font color="Green">E266</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">U</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E16:117_3"/><i><font color="Green">E267</font></i>: 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">d</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:117_3"/><i><font color="Green">E268</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="toprealb.html#T19" target="_self">Th19</a>, , </i>;<br/>
<a NAME="E18:117_3"/><i><font color="Green">E269</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="toprealb.html#T19" target="_self">Th19</a>, <a class="ref" href="topreala.html#T29">TOPREALA:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">J</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">p1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> 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="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">J</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">J</font></span>)</span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E8"><i><font color="Green">Lemma51</font></i></a></i>;<br/>
<a NAME="E21:117_3"/>
<font color="Maroon">d</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">J</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">d2</font> = <font color="Maroon">d</font> as   <a href="struct_0.html#NM2">Subset</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">J</font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E23:117_3"/><i><font color="Green">E270</font></i>: 
<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">J</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a> <a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<a NAME="E24:117_3"/><i><font color="Green">E271</font></i>: 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">J</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">d1</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>;<br/>
<a NAME="E25:117_3"/><i><font color="Green">E272</font></i>: 
<font color="Maroon">f1</font> is <a href="funct_2.html#V2">onto</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_3_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:117_3_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U1</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>,<a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a><br/>

<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="E2:117_3_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U1</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E4:117_3_2"/><i><font color="Green">E273</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E5:117_3_2"/><i><font color="Green">E274</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="toprealb.html#T14" target="_self">Th14</a>, <a class="ref" href="toprealb.html#T15" target="_self">Th15</a>, , , <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E6:117_3_2"/><i><font color="Green">E275</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E6"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E7:117_3_2"/>
<span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E8:117_3_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="ref" href="toprealb.html#T19" target="_self">Th19</a>, , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">nn</font> = <font color="Maroon">n</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>, <a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E27:117_3"/>
 <a href="toprealb.html#K13" target="_self">CircleMap</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4" target="_self">R^1</a> <font color="Maroon">n</font></span>)</span> is <a href="t_0topsp.html#V1">open</a>
 
;<br/>
<a NAME="E28:117_3"/><b>then </b><i><font color="Green">E293</font></i>: 
<font color="Maroon">F</font> is <a href="t_0topsp.html#V1">open</a>
 
<b>by </b><i><a class="ref" href="topreala.html#T33">TOPREALA:33</a></i>;<br/>
<a NAME="E29:117_3"/>
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">n</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> is <a href="funct_1.html#V2">one-to-one</a>
 
;<br/>
<a NAME="E30:117_3"/><b>then </b>
<font color="Maroon">f</font> is <a href="t_0topsp.html#V1">open</a>
 
<b>by </b><i><a class="ref" href="toprealb.html#T19" target="_self">Th19</a>, , , , , <a class="ref" href="topreala.html#T31">TOPREALA:31</a></i>;<br/>
<b>hence </b><a NAME="E31:117_3"/>
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, , , <a class="ref" href="topreala.html#T37">TOPREALA:37</a></i>;<br/>


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

<b>assume </b><a NAME="E11:117"/><i><font color="Green">E304</font></i>: 
<font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="topalg_2.html#K3">R^1</a> , <font color="Maroon">x2</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/><b>assume </b><a NAME="E1:117_4"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:117_4"/><i><font color="Green">E305</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E4:117_4"/><i><font color="Green">E306</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E6:117_4"/><i><font color="Green">E307</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i/>;<br/><b>assume </b><a NAME="E7:117_4"/><i><font color="Green">E308</font></i>: 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font>
 ;<br/><a NAME="E8:117_4"/><i><font color="Green">E309</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="ref" href="toprealb.html#T15" target="_self">Th15</a>, <a class="ref" href="toprealb.html#T19" target="_self">Th19</a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E9:117_4"/>
0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E10:117_4"/><b>then </b><i><font color="Green">E310</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:117_4"/><i><font color="Green">E311</font></i>: 
<font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="toprealb.html#T15" target="_self">Th15</a>, <a class="ref" href="toprealb.html#T19" target="_self">Th19</a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E12:117_4"/><b>then </b><i><font color="Green">E312</font></i>: 
<font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 3 <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><b>set </b><font color="Maroon">k</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>;<br/><a NAME="E13:117_4"/><i><font color="Green">E313</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:117_4_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> or <font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> or <font color="Maroon">x1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_4_1_1"><b>suppose </b></a><a NAME="E1:117_4_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:117_4_1_1"/><b>then </b>
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="toprealb.html#K9" target="_self">c[10]</a> 
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">x2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="txt" href="toprealb.html#E34"><i><font color="Green">Lemma104</font></i></a>, <a class="txt" href="toprealb.html#E6"><i><font color="Green">Lemma43</font></i></a>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E4:117_4_1_1"/>
<font color="Maroon">w</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><a NAME="E5:117_4_1_1"/><b>then </b><i><font color="Green">E314</font></i>: 
<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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R2">in</a>  <a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span>
 ;<br/><a NAME="E6:117_4_1_1"/><i><font color="Green">E315</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E7:117_4_1_1"/>
0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E8:117_4_1_1"/><b>then </b>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E9:117_4_1_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_4_1_2"><b>suppose </b></a><a NAME="E1:117_4_1_2"/><i><font color="Green">E316</font></i>: 
<font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">K</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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/><a NAME="E2:117_4_1_2"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><a NAME="E3:117_4_1_2"/><b>then </b><i><font color="Green">E317</font></i>: 
<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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R2">in</a>  <a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span>
 ;<br/><a NAME="E4:117_4_1_2"/><i><font color="Green">E318</font></i>: 
( <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E5:117_4_1_2"/>
( <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K20">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p5"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K20">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p5"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="txt" href="toprealb.html#E5"><i><font color="Green">Lemma42</font></i></a></i>;<br/><a NAME="E6:117_4_1_2"/><b>then </b><i><font color="Green">E319</font></i>: 
(  <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> &amp;  <a href="sin_cos.html#K20">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K20">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="sppol_2.html#T1">SPPOL_2:1</a></i>;<br/><a NAME="E7:117_4_1_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/><a NAME="E8:117_4_1_2"/><b>then </b><i><font color="Green">E320</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E9:117_4_1_2"/>
<font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E10:117_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E11:117_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E12:117_4_1_2"/><b>then </b><i><font color="Green">E321</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E13:117_4_1_2"/>
<font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E14:117_4_1_2"/><b>then </b><i><font color="Green">E322</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E15:117_4_1_2"/>
<font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><a NAME="E16:117_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E17:117_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E18:117_4_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E19:117_4_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="complex2.html#T12">COMPLEX2:12</a></i>;<br/><a NAME="E20:117_4_1_2"/><b>then </b><i><font color="Green">E323</font></i>: 
<font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E21:117_4_1_2"/><b>then </b><i><font color="Green">E324</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 ;<br/><a NAME="E22:117_4_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E23:117_4_1_2"/><b>then </b><i><font color="Green">E325</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E24:117_4_1_2"/>
<font color="Maroon">x2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E25:117_4_1_2"/><b>then </b>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E26:117_4_1_2"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_4_1_3"><b>suppose </b></a><a NAME="E1:117_4_1_3"/><i><font color="Green">E326</font></i>: 
<font color="Maroon">x1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">K</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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/><a NAME="E2:117_4_1_3"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><a NAME="E3:117_4_1_3"/><b>then </b><i><font color="Green">E327</font></i>: 
<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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R2">in</a>  <a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span>
 ;<br/><a NAME="E4:117_4_1_3"/><i><font color="Green">E328</font></i>: 
( <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E5:117_4_1_3"/>
( <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K20">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p5"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K20">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p5"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="txt" href="toprealb.html#E5"><i><font color="Green">Lemma42</font></i></a></i>;<br/><a NAME="E6:117_4_1_3"/><b>then </b><i><font color="Green">E329</font></i>: 
(  <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> &amp;  <a href="sin_cos.html#K20">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K20">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="sppol_2.html#T1">SPPOL_2:1</a></i>;<br/><a NAME="E7:117_4_1_3"/><i><font color="Green">E330</font></i>: 
<font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:117_4_1_3"/><i><font color="Green">E331</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E9:117_4_1_3"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E10:117_4_1_3"/><b>then </b><i><font color="Green">E332</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E11:117_4_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E12:117_4_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</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/><a NAME="E13:117_4_1_3"/><b>then </b><i><font color="Green">E333</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E14:117_4_1_3"/>
<font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E15:117_4_1_3"/><b>then </b><i><font color="Green">E334</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E16:117_4_1_3"/>
<font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><a NAME="E17:117_4_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E18:117_4_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E19:117_4_1_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E20:117_4_1_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="complex2.html#T12">COMPLEX2:12</a></i>;<br/><a NAME="E21:117_4_1_3"/><b>then </b><i><font color="Green">E335</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E22:117_4_1_3"/><b>then </b><i><font color="Green">E336</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 ;<br/><a NAME="E23:117_4_1_3"/><i><font color="Green">E337</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E24:117_4_1_3"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><a NAME="E25:117_4_1_3"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E26:117_4_1_3"/><b>then </b><i><font color="Green">E338</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E27:117_4_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E28:117_4_1_3"/><b>then </b>
<font color="Maroon">x2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E29:117_4_1_3"/><b>then </b>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x2</font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E30:117_4_1_3"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:117"/>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">U</font>
 <b>by </b><i><a class="ref" href="toprealb.html#T13" target="_self">Th13</a>, <a class="ref" href="toprealb.html#T14" target="_self">Th14</a>, <a class="ref" href="t_0topsp.html#T2">T_0TOPSP:2</a></i>;<br/>

<b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/>

<b>assume </b><a NAME="E14:117"/><i><font color="Green">E339</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <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">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U</font></span>)</span>;<br/>

<b>assume </b><a NAME="E15:117"/><i><font color="Green">E340</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">d</font>
 ;<br/>

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E17:117"/><i><font color="Green">E341</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="toprealb.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E18:117"/>
( <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> &amp; 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">d</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="toprealb.html#T19" target="_self">Th19</a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E20:117"/><i><font color="Green">E342</font></i>: 
1 <a href="hidden.html#R2">in</a> <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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<b>reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">U</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="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> <b>by </b><i><a class="ref" href="toprealb.html#T14" target="_self">Th14</a>, , <a class="txt" href="toprealb.html#E6"><i><font color="Green">Lemma43</font></i></a>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</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">d1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U1</font></span>)</span> ;<br/>
<a NAME="E23:117"/><i><font color="Green">E343</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">U</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E24:117"/><i><font color="Green">E344</font></i>: 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" target="_self">IntIntervals</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="toprealb.html#T15" target="_self">Th15</a>, </i>;<br/>
<a NAME="E25:117"/><i><font color="Green">E345</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="toprealb.html#T15" target="_self">Th15</a>, , , </i>;<br/>
<a NAME="E26:117"/><i><font color="Green">E346</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="topreala.html#T29">TOPREALA:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">J</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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">p1</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> 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="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">J</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">J</font></span>)</span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E8"><i><font color="Green">Lemma51</font></i></a></i>;<br/>
<a NAME="E29:117"/>
<font color="Maroon">d</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">J</font>
 
<b>by </b><i><a class="ref" href="toprealb.html#T19" target="_self">Th19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">d2</font> = <font color="Maroon">d</font> as   <a href="struct_0.html#NM2">Subset</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">J</font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E31:117"/><i><font color="Green">E347</font></i>: 
<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">J</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a> <a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="toprealb.html#T19" target="_self">Th19</a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<a NAME="E32:117"/><i><font color="Green">E348</font></i>: 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">J</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">d1</font>
 
<b>by </b><i><a class="ref" href="toprealb.html#T19" target="_self">Th19</a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>;<br/>
<a NAME="E33:117"/><i><font color="Green">E349</font></i>: 
<font color="Maroon">f1</font> is <a href="funct_2.html#V2">onto</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:117_5"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U1</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>,<a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a><br/>

<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="E2:117_5"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">U1</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E4:117_5"/><i><font color="Green">E350</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E5:117_5"/><i><font color="Green">E351</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="toprealb.html#T13" target="_self">Th13</a>, <a class="ref" href="toprealb.html#T14" target="_self">Th14</a>, , , <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E6:117_5"/><i><font color="Green">E352</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E6"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E7:117_5"/>
<span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E8:117_5"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">nn</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</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>, <a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E35:117"/>
 <a href="toprealb.html#K13" target="_self">CircleMap</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4" target="_self">R^1</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> is <a href="t_0topsp.html#V1">open</a>
 
;<br/>
<a NAME="E36:117"/><b>then </b><i><font color="Green">E353</font></i>: 
<font color="Maroon">F</font> is <a href="t_0topsp.html#V1">open</a>
 
<b>by </b><i><a class="ref" href="topreala.html#T33">TOPREALA:33</a></i>;<br/>
<a NAME="E37:117"/>
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <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">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> is <a href="funct_1.html#V2">one-to-one</a>
 
;<br/>
<a NAME="E38:117"/><b>then </b>
<font color="Maroon">f</font> is <a href="t_0topsp.html#V1">open</a>
 
<b>by </b><i>, , , , , <a class="ref" href="topreala.html#T31">TOPREALA:31</a></i>;<br/>
<b>hence </b><a NAME="E39:117"/>
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, , , <a class="ref" href="topreala.html#T37">TOPREALA:37</a></i>;<br/>


</div>
