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

<b>set </b><font color="Maroon">c<sub>56</sub></font> =  <a href="toprealb.html#K3" target="_self">IntIntervals</a> 0,1;<br/>
<b>set </b><font color="Maroon">c<sub>57</sub></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="E1:121"/><i><font color="Green">E104</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">c<sub>58</sub></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">c<sub>59</sub></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">c<sub>60</sub></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">c<sub>61</sub></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">c<sub>61</sub></font>
;<br/>

<a NAME="E3:121"/><i><font color="Green">E105</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><a class="txt" href="toprealb.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E4:121"/><b>then </b><i><font color="Green">E106</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><a class="ref" href="toprealb.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E5:121"/>
<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><a class="txt" href="toprealb.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E6:121"/><b>then </b><i><font color="Green">E107</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><a class="ref" href="toprealb.html#T41" target="_self">Th41</a></i>;<br/>
<b>hence </b><a NAME="E7:121"/>
<font color="Maroon">c<sub>61</sub></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="txt" href="toprealb.html#E3:121"><i><font color="Green">E105</font></i></a>, <a class="ref" href="toprealb.html#T41" target="_self">Th41</a></i>;<br/>

<b>thus </b><a NAME="E8:121"/>
<font color="Maroon">c<sub>61</sub></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="121_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:121_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">c<sub>61</sub></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="121_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>62</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:121_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>62</sub></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">c<sub>63</sub></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">c<sub>64</sub></font> = <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>63</sub></font> as   <a href="struct_0.html#NM7">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">c<sub>63</sub></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#E73"><i><font color="Green">Lemma51</font></i></a></i>;<br/>
<a NAME="E4:121_1_1"/>
<font color="Maroon">c<sub>64</sub></font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="ref" href="toprealb.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E5:121_1_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>64</sub></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">c<sub>65</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:121_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>65</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>64</sub></font>
 <b>and </b><br/><a NAME="E8:121_1_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>64</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>65</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>62</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E1:121_1_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:121_1_1"/><i><font color="Green">E111</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>64</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>63</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E62"><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">c<sub>66</sub></font> = <font color="Maroon">c<sub>65</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="toprealb.html#E7:121_1_1"><i><font color="Green">E109</font></i></a></i>;<br/>
<a NAME="E11:121_1_1"/><i><font color="Green">E112</font></i>: 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>66</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>64</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>66</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E7:121_1_1"><i><font color="Green">E109</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E12:121_1_1"/><i><font color="Green">E113</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">c<sub>61</sub></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="121_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:121_1_1_1"/>
( <font color="Maroon">c<sub>66</sub></font> <a href="hidden.html#R1">=</a> 0 or ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>66</sub></font> &amp; <font color="Maroon">c<sub>66</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) or ( <font color="Maroon">c<sub>66</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>66</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="toprealb.html#E7:121_1_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="toprealb.html#E9:121_1_1"><i><font color="Green">E111</font></i></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="121_1_1_1_1"><b>suppose </b></a><a NAME="E1:121_1_1_1_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>66</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:121_1_1_1_1"/>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>63</sub></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">E115</font></i>: <font color="Maroon">c<sub>64</sub></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><a class="ref" href="toprealb.html#T33" target="_self">Th33</a></i>
<br/>.= 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="toprealb.html#T33" target="_self">Th33</a></i>
;<br/><a NAME="E4:121_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="E5:121_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>62</sub></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#E8:121_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="toprealb.html#E1:121_1_1_1_1"><i><font color="Green">E114</font></i></a>, <a class="txt" href="toprealb.html#E3:121_1_1_1_1"><i><font color="Green">E115</font></i></a>, <a class="txt" href="toprealb.html#E62"><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="E6:121_1_1_1_1"/>
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>61</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E6:121"><i><font color="Green">E107</font></i></a>, <a class="txt" href="toprealb.html#E12:121_1_1"><i><font color="Green">E113</font></i></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="121_1_1_1_2"><b>suppose </b></a><a NAME="E1:121_1_1_1_2"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>66</sub></font> &amp; <font color="Maroon">c<sub>66</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:121_1_1_1_2"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>66</sub></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="E3:121_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><a class="txt" href="toprealb.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/><a NAME="E4:121_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>66</sub></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="txt" href="toprealb.html#E2:121_1_1_1_2"><i><font color="Green">E114</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E5:121_1_1_1_2"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>62</sub></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#E8:121_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="toprealb.html#E11:121_1_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="toprealb.html#E62"><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="E6:121_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">c<sub>61</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:121_1_1_1_2"/>
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>61</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E5:121_1_1_1_2"><i><font color="Green">E115</font></i></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="121_1_1_1_3"><b>suppose </b></a><a NAME="E1:121_1_1_1_3"/>
( <font color="Maroon">c<sub>66</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>66</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:121_1_1_1_3"/>
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>61</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E2:121_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>61</sub></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="E9:121"/>
<font color="Maroon">c<sub>61</sub></font> is <a href="tops_2.html#V1">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>62</sub></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/>

<i><font color="Green">E108</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> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p3">(<span class="default">0 <a href="real_1.html#K3">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></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/>.= 
<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>

;<br/>
<i><font color="Green">E109</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> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<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="real_1.html#K3">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></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/>.= 
<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>

;<br/>
<b>reconsider </b><font color="Maroon">c<sub>63</sub></font> = 0 as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<div><i><font color="Green">E110</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="121_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>64</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><i><font color="Green">E111</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">c<sub>64</sub></font></span>)</span></span>)</span> = 
 <a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">c<sub>64</sub></font>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
<br/>.= 
<font color="Maroon">c<sub>64</sub></font>

;<br/><a NAME="E2:121_2_3"/>
<font color="Maroon">c<sub>64</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>64</sub></font>
 ;<br/><b>hence </b><a NAME="E3:121_2_3"/>
<font color="Maroon">c<sub>64</sub></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">c<sub>64</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E1:121_2_3"><i><font color="Green">E111</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>64</sub></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">c<sub>63</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>63</sub></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">c<sub>65</sub></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">c<sub>63</sub></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">c<sub>63</sub></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><a class="txt" href="toprealb.html#E4:121_2"><i><font color="Green">E110</font></i></a></i>;<br/>
<div><i><font color="Green">E111</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="121_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>66</sub></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">c<sub>67</sub></font> = <font color="Maroon">c<sub>66</sub></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">c<sub>66</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E4:121_2"><i><font color="Green">E110</font></i></a></i>;<br/><i><font color="Green">E112</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">c<sub>66</sub></font></span>)</span></span>)</span> = 
 <a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">c<sub>66</sub></font>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
<br/>.= 
<font color="Maroon">c<sub>66</sub></font>

;<br/><font color="Maroon">c<sub>67</sub></font> = 
<font color="Maroon">c<sub>67</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>66</sub></font>

<br/>.= 
<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">c<sub>66</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>66</sub></font>
<b>by </b><i><a class="txt" href="toprealb.html#E2:121_2_4"><i><font color="Green">E112</font></i></a></i>
<br/>.= 
<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">c<sub>66</sub></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">c<sub>66</sub></font></span>)</span>

;<br/><b>hence </b><a NAME="E4:121_2_4"/>
<font color="Maroon">c<sub>66</sub></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">c<sub>66</sub></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="E8:121_2"/><b>then </b>
<font color="Maroon">c<sub>64</sub></font> is <a href="pre_topc.html#V3">open</a>
 
;<br/>
<a NAME="E9:121_2"/><b>then </b><i><font color="Green">E112</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">c<sub>63</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>64</sub></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="E10:121_2"/>
<font color="Maroon">c<sub>65</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E7:121_2"><i><font color="Green">E111</font></i></a></i>;<br/>
<a NAME="E11:121_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">c<sub>63</sub></font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>65</sub></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="E12:121_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="txt" href="toprealb.html#E4:121"><i><font color="Green">E106</font></i></a>, <a class="txt" href="toprealb.html#E6:121"><i><font color="Green">E107</font></i></a>, <a class="txt" href="toprealb.html#E1:121_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="toprealb.html#E2:121_2"><i><font color="Green">E109</font></i></a>, <a class="txt" href="toprealb.html#E9:121_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="tsep_1.html#T17">TSEP_1:17</a></i>;<br/>
<b>hence </b><a NAME="E13:121_2"/>
(  not <font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>61</sub></font> or <font color="Maroon">c<sub>62</sub></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">c<sub>62</sub></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="E10:121"/>
( <font color="Maroon">c<sub>62</sub></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">c<sub>62</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  st <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></font> being  <a href="struct_0.html#NM7">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">b<sub>1</sub></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">c<sub>62</sub></font></span>)</span> st <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:121_3"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>62</sub></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="E2:121_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><a class="txt" href="toprealb.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E3:121_3"/><b>then </b><i><font color="Green">E109</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><a class="ref" href="toprealb.html#T41" target="_self">Th41</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>63</sub></font>, <font color="Maroon">c<sub>64</sub></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="E1:121_3_1"/>
<font color="Maroon">c<sub>63</sub></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">c<sub>65</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:121_3_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>63</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>65</sub></font>
 <b>and </b><br/><a NAME="E4:121_3_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>65</sub></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">c<sub>66</sub></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="E6:121_3_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>65</sub></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">c<sub>66</sub></font></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>66</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E4:121_3_1"><i><font color="Green">E111</font></i></a></i>;<br/><b>assume </b><a NAME="E7:121_3_1"/><i><font color="Green">E113</font></i>: 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>63</sub></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">c<sub>64</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>67</sub></font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span>;<br/><b>set </b><font color="Maroon">c<sub>68</sub></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">c<sub>64</sub></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">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/><a NAME="E8:121_3_1"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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="E9:121_3_1"/><b>then </b><i><font color="Green">E114</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">c<sub>64</sub></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">c<sub>64</sub></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="E10:121_3_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>66</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>63</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E3:121_3_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="toprealb.html#E6:121_3_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E11:121_3_1"/>
<font color="Maroon">c<sub>63</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>66</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E3:121_3_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="toprealb.html#E6:121_3_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E12:121_3_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>63</sub></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">c<sub>66</sub></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="E13:121_3_1"/><b>then </b><i><font color="Green">E117</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>63</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>66</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E10:121_3_1"><i><font color="Green">E115</font></i></a>, <a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><a NAME="E14:121_3_1"/><i><font color="Green">E118</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>64</sub></font>
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><a NAME="E15:121_3_1"/>
not <font color="Maroon">c<sub>63</sub></font> is <a href="int_1.html#V1">integer</a>
 <b>by </b><i><a class="txt" href="toprealb.html#E10:121_3_1"><i><font color="Green">E115</font></i></a>, <a class="txt" href="toprealb.html#E13:121_3_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="int_1.html#T48">INT_1:48</a></i>;<br/><a NAME="E16:121_3_1"/><b>then </b><i><font color="Green">E119</font></i>: 
not <font color="Maroon">c<sub>63</sub></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="121_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:121_3_1_1"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>64</sub></font>
 ;<br/><a NAME="E2:121_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">c<sub>63</sub></font> <a href="hidden.html#R1">=</a>  <a href="toprealb.html#K9" target="_self">c[10]</a> 
 <b>by </b><i><a class="txt" href="toprealb.html#E7:121_3_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="toprealb.html#T33" target="_self">Th33</a></i>;<br/><b>hence </b><a NAME="E3:121_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="toprealb.html#E1:121"><i><font color="Green">E104</font></i></a>, <a class="txt" href="toprealb.html#E16:121_3_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="toprealb.html#E62"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="toprealb.html#T34" target="_self">Th34</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="E18:121_3_1"/><b>then </b><i><font color="Green">E120</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>64</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E14:121_3_1"><i><font color="Green">E118</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E19:121_3_1"/><i><font color="Green">E121</font></i>: 
( <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>63</sub></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">c<sub>63</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>66</sub></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">c<sub>64</sub></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">c<sub>64</sub></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">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="toprealb.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E20:121_3_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>66</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E10:121_3_1"><i><font color="Green">E115</font></i></a>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/><a NAME="E21:121_3_1"/><b>then </b><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E22:121_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>66</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E12:121_3_1"><i><font color="Green">E116</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E23:121_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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="E24:121_3_1"/><b>then </b><i><font color="Green">E123</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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#K33">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="E25:121_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>64</sub></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">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E18:121_3_1"><i><font color="Green">E120</font></i></a>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/><a NAME="E26:121_3_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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="E27:121_3_1"/>
<font color="Maroon">c<sub>64</sub></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">c<sub>64</sub></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="E28:121_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>64</sub></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">c<sub>64</sub></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">c<sub>64</sub></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">c<sub>64</sub></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="E29:121_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>64</sub></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">c<sub>64</sub></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="E30:121_3_1"/><b>then </b><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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#K33">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="E31:121_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">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p5"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p5"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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="ref" href="toprealb.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E32:121_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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="toprealb.html#E7:121_3_1"><i><font color="Green">E113</font></i></a>, <a class="txt" href="toprealb.html#E19:121_3_1"><i><font color="Green">E121</font></i></a>, <a class="ref" href="sppol_2.html#T1">SPPOL_2:1</a></i>;<br/><a NAME="E33:121_3_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E21:121_3_1"><i><font color="Green">E122</font></i></a>, <a class="txt" href="toprealb.html#E24:121_3_1"><i><font color="Green">E123</font></i></a>, <a class="txt" href="toprealb.html#E26:121_3_1"><i><font color="Green">E124</font></i></a>, <a class="txt" href="toprealb.html#E30:121_3_1"><i><font color="Green">E125</font></i></a>, <a class="ref" href="complex2.html#T12">COMPLEX2:12</a></i>;<br/><a NAME="E34:121_3_1"/><b>then </b>
<font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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="E35:121_3_1"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>64</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span>
 ;<br/><a NAME="E36:121_3_1"/>
<font color="Maroon">c<sub>63</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>66</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E3:121_3_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="toprealb.html#E6:121_3_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E37:121_3_1"/><b>then </b>
<font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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="E38:121_3_1"/><b>then </b>
<font color="Maroon">c<sub>64</sub></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">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E35:121_3_1"><i><font color="Green">E126</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E39:121_3_1"/><b>then </b>
<font color="Maroon">c<sub>64</sub></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">c<sub>64</sub></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">c<sub>64</sub></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="txt" href="toprealb.html#E18:121_3_1"><i><font color="Green">E120</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E40:121_3_1"/>
<font color="Maroon">c<sub>64</sub></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="txt" href="toprealb.html#E9:121_3_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:121_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">c<sub>62</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E1:121_3"><i><font color="Green">E108</font></i></a>, <a class="txt" href="toprealb.html#E3:121_3"><i><font color="Green">E109</font></i></a>, <a class="ref" href="t_0topsp.html#T2">T_0TOPSP:2</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>63</sub></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="E6:121_3"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>63</sub></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">c<sub>64</sub></font> be   <a href="struct_0.html#NM7">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">c<sub>63</sub></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">c<sub>62</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E7:121_3"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>64</sub></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">c<sub>63</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>65</sub></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="E9:121_3"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>63</sub></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">c<sub>65</sub></font></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>65</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E6:121_3"><i><font color="Green">E110</font></i></a></i>;<br/>
<a NAME="E10:121_3"/>
( 0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>65</sub></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">c<sub>65</sub></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">c<sub>65</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>65</sub></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">c<sub>66</sub></font> = <font color="Maroon">c<sub>63</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="toprealb.html#E9:121_3"><i><font color="Green">E112</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E12:121_3"/><i><font color="Green">E113</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">c<sub>67</sub></font> = <font color="Maroon">c<sub>62</sub></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="txt" href="toprealb.html#E1:121_3"><i><font color="Green">E108</font></i></a>, <a class="txt" href="toprealb.html#E12:121_3"><i><font color="Green">E113</font></i></a>, <a class="txt" href="toprealb.html#E62"><i><font color="Green">Lemma43</font></i></a>, <a class="txt" href="toprealb.html#E66"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>68</sub></font> = <font color="Maroon">c<sub>64</sub></font> as   <a href="struct_0.html#NM7">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">c<sub>66</sub></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">c<sub>67</sub></font></span>)</span> ;<br/>
<a NAME="E15:121_3"/><i><font color="Green">E114</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">c<sub>62</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>62</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E16:121_3"/><i><font color="Green">E115</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">c<sub>63</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E6:121_3"><i><font color="Green">E110</font></i></a>, <a class="ref" href="toprealb.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E17:121_3"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>64</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E6:121_3"><i><font color="Green">E110</font></i></a>, <a class="txt" href="toprealb.html#E7:121_3"><i><font color="Green">E111</font></i></a>, <a class="txt" href="toprealb.html#E5"><i><font color="Green">Lemma3</font></i></a>, <a class="ref" href="toprealb.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E18:121_3"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">c<sub>64</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E7:121_3"><i><font color="Green">E111</font></i></a>, <a class="ref" href="topreala.html#T29">TOPREALA:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>69</sub></font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>65</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>65</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>5</sub></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">c<sub>70</sub></font> = <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>69</sub></font> as   <a href="struct_0.html#NM7">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">c<sub>69</sub></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#E73"><i><font color="Green">Lemma51</font></i></a></i>;<br/>
<a NAME="E21:121_3"/>
<font color="Maroon">c<sub>63</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>69</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E9:121_3"><i><font color="Green">E112</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>71</sub></font> = <font color="Maroon">c<sub>63</sub></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">c<sub>69</sub></font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E23:121_3"/><i><font color="Green">E118</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">c<sub>69</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>71</sub></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">c<sub>63</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E9:121_3"><i><font color="Green">E112</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<a NAME="E24:121_3"/><i><font color="Green">E119</font></i>: 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>63</sub></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">c<sub>69</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>66</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E9:121_3"><i><font color="Green">E112</font></i></a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>;<br/>
<a NAME="E25:121_3"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>68</sub></font> is <a href="funct_2.html#V2">onto</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:121_3_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>68</sub></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">c<sub>67</sub></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">c<sub>72</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:121_3_2"/>
<font color="Maroon">c<sub>72</sub></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">c<sub>67</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>73</sub></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="E4:121_3_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>73</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>63</sub></font>
 <b>and </b><br/><a NAME="E5:121_3_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">c<sub>72</sub></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">c<sub>73</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E1:121_3"><i><font color="Green">E108</font></i></a>, <a class="txt" href="toprealb.html#E3:121_3"><i><font color="Green">E109</font></i></a>, <a class="txt" href="toprealb.html#E15:121_3"><i><font color="Green">E114</font></i></a>, <a class="txt" href="toprealb.html#E16:121_3"><i><font color="Green">E115</font></i></a>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E6:121_3_2"/><i><font color="Green">E123</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">c<sub>63</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>63</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E62"><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="E7:121_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">c<sub>63</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>73</sub></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">c<sub>73</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E4:121_3_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E8:121_3_2"/>
<font color="Maroon">c<sub>72</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>68</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E7:121_3"><i><font color="Green">E111</font></i></a>, <a class="txt" href="toprealb.html#E4:121_3_2"><i><font color="Green">E121</font></i></a>, <a class="txt" href="toprealb.html#E5:121_3_2"><i><font color="Green">E122</font></i></a>, <a class="txt" href="toprealb.html#E6:121_3_2"><i><font color="Green">E123</font></i></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">c<sub>72</sub></font> = <font color="Maroon">c<sub>65</sub></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="E27:121_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">c<sub>65</sub></font></span>)</span> is <a href="t_0topsp.html#V1">open</a>
 
;<br/>
<a NAME="E28:121_3"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>70</sub></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="E29:121_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">c<sub>65</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>65</sub></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="E30:121_3"/><b>then </b>
<font color="Maroon">c<sub>64</sub></font> is <a href="t_0topsp.html#V1">open</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E7:121_3"><i><font color="Green">E111</font></i></a>, <a class="txt" href="toprealb.html#E23:121_3"><i><font color="Green">E118</font></i></a>, <a class="txt" href="toprealb.html#E24:121_3"><i><font color="Green">E119</font></i></a>, <a class="txt" href="toprealb.html#E25:121_3"><i><font color="Green">E120</font></i></a>, <a class="txt" href="toprealb.html#E28:121_3"><i><font color="Green">E121</font></i></a>, <a class="ref" href="topreala.html#T31">TOPREALA:31</a></i>;<br/>
<b>hence </b><a NAME="E31:121_3"/>
<font color="Maroon">c<sub>64</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="toprealb.html#E17:121_3"><i><font color="Green">E116</font></i></a>, <a class="txt" href="toprealb.html#E18:121_3"><i><font color="Green">E117</font></i></a>, <a class="txt" href="toprealb.html#E25:121_3"><i><font color="Green">E120</font></i></a>, <a class="ref" href="topreala.html#T37">TOPREALA:37</a></i>;<br/>


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

<b>assume </b><a NAME="E11:121"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>62</sub></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="121_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>63</sub></font>, <font color="Maroon">c<sub>64</sub></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="E1:121_4"/>
<font color="Maroon">c<sub>63</sub></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">c<sub>65</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:121_4"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>63</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>65</sub></font>
 <b>and </b><br/><a NAME="E4:121_4"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>65</sub></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">c<sub>66</sub></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="E6:121_4"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>65</sub></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">c<sub>66</sub></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">c<sub>66</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E4:121_4"><i><font color="Green">E110</font></i></a></i>;<br/><b>assume </b><a NAME="E7:121_4"/><i><font color="Green">E112</font></i>: 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>63</sub></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">c<sub>64</sub></font>
 ;<br/><a NAME="E8:121_4"/><i><font color="Green">E113</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">c<sub>66</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>63</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E3:121_4"><i><font color="Green">E109</font></i></a>, <a class="txt" href="toprealb.html#E6:121_4"><i><font color="Green">E111</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E9:121_4"/>
0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>66</sub></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">c<sub>66</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E10:121_4"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>66</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>63</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E8:121_4"><i><font color="Green">E113</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:121_4"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>63</sub></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">c<sub>66</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E3:121_4"><i><font color="Green">E109</font></i></a>, <a class="txt" href="toprealb.html#E6:121_4"><i><font color="Green">E111</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E12:121_4"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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">c<sub>67</sub></font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span>;<br/><a NAME="E13:121_4"/><i><font color="Green">E117</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>64</sub></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="121_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:121_4_1"/>
( <font color="Maroon">c<sub>63</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>66</sub></font> or <font color="Maroon">c<sub>63</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>66</sub></font> or <font color="Maroon">c<sub>63</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>66</sub></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="121_4_1_1"><b>suppose </b></a><a NAME="E1:121_4_1_1"/>
<font color="Maroon">c<sub>63</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>66</sub></font>
 ;<br/><div class="add"><a NAME="E2:121_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">c<sub>64</sub></font> <a href="hidden.html#R1">=</a>  <a href="toprealb.html#K9" target="_self">c[10]</a> 
 <b>by </b><i><a class="txt" href="toprealb.html#E7:121_4"><i><font color="Green">E112</font></i></a>, <a class="ref" href="toprealb.html#T33" target="_self">Th33</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>68</sub></font> = <font color="Maroon">c<sub>64</sub></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#E1:121"><i><font color="Green">E104</font></i></a>, <a class="txt" href="toprealb.html#E62"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="toprealb.html#T34" target="_self">Th34</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="E4:121_4_1_1"/>
<font color="Maroon">c<sub>68</sub></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="E5:121_4_1_1"/><b>then </b><i><font color="Green">E118</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">c<sub>68</sub></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">c<sub>68</sub></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="E6:121_4_1_1"/><i><font color="Green">E119</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">c<sub>68</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>68</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E7:121_4_1_1"/>
0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>68</sub></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">c<sub>68</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E8:121_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>64</sub></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">c<sub>68</sub></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">c<sub>68</sub></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="txt" href="toprealb.html#E6:121_4_1_1"><i><font color="Green">E119</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E9:121_4_1_1"/>
<font color="Maroon">c<sub>64</sub></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="txt" href="toprealb.html#E5:121_4_1_1"><i><font color="Green">E118</font></i></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="121_4_1_2"><b>suppose </b></a><a NAME="E1:121_4_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>63</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>66</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>68</sub></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">c<sub>64</sub></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">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/><a NAME="E2:121_4_1_2"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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="E3:121_4_1_2"/><b>then </b><i><font color="Green">E119</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">c<sub>64</sub></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">c<sub>64</sub></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="E4:121_4_1_2"/><i><font color="Green">E120</font></i>: 
( <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>63</sub></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">c<sub>63</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>66</sub></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">c<sub>64</sub></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">c<sub>64</sub></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">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="toprealb.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E5:121_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">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p5"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p5"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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="ref" href="toprealb.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E6:121_4_1_2"/><b>then </b><i><font color="Green">E121</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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="toprealb.html#E7:121_4"><i><font color="Green">E112</font></i></a>, <a class="txt" href="toprealb.html#E4:121_4_1_2"><i><font color="Green">E120</font></i></a>, <a class="ref" href="sppol_2.html#T1">SPPOL_2:1</a></i>;<br/><a NAME="E7:121_4_1_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>66</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E10:121_4"><i><font color="Green">E114</font></i></a>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/><a NAME="E8:121_4_1_2"/><b>then </b><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E9:121_4_1_2"/>
<font color="Maroon">c<sub>63</sub></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">c<sub>66</sub></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="txt" href="toprealb.html#E1:121_4_1_2"><i><font color="Green">E118</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E10:121_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>66</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E11:121_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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="E12:121_4_1_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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#K33">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="E13:121_4_1_2"/>
<font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>64</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E13:121_4"><i><font color="Green">E117</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E14:121_4_1_2"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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="E15:121_4_1_2"/>
<font color="Maroon">c<sub>64</sub></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">c<sub>64</sub></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="E16:121_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>64</sub></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">c<sub>64</sub></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">c<sub>64</sub></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">c<sub>64</sub></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="E17:121_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>64</sub></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">c<sub>64</sub></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="E18:121_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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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#K33">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="E19:121_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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E6:121_4_1_2"><i><font color="Green">E121</font></i></a>, <a class="txt" href="toprealb.html#E8:121_4_1_2"><i><font color="Green">E122</font></i></a>, <a class="txt" href="toprealb.html#E12:121_4_1_2"><i><font color="Green">E123</font></i></a>, <a class="txt" href="toprealb.html#E14:121_4_1_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="complex2.html#T12">COMPLEX2:12</a></i>;<br/><a NAME="E20:121_4_1_2"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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="E21:121_4_1_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>64</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span>
 ;<br/><a NAME="E22:121_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">c<sub>66</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E8:121_4"><i><font color="Green">E113</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E23:121_4_1_2"/><b>then </b><i><font color="Green">E127</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">c<sub>64</sub></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">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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">c<sub>64</sub></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="E24:121_4_1_2"/>
<font color="Maroon">c<sub>64</sub></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">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E12:121_4"><i><font color="Green">E116</font></i></a>, <a class="txt" href="toprealb.html#E21:121_4_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E25:121_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>64</sub></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">c<sub>64</sub></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">c<sub>64</sub></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="txt" href="toprealb.html#E20:121_4_1_2"><i><font color="Green">E125</font></i></a>, <a class="txt" href="toprealb.html#E23:121_4_1_2"><i><font color="Green">E127</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E26:121_4_1_2"/>
<font color="Maroon">c<sub>64</sub></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="txt" href="toprealb.html#E3:121_4_1_2"><i><font color="Green">E119</font></i></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="121_4_1_3"><b>suppose </b></a><a NAME="E1:121_4_1_3"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>63</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>66</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>68</sub></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">c<sub>64</sub></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">c<sub>64</sub></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="E2:121_4_1_3"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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="E3:121_4_1_3"/><b>then </b><i><font color="Green">E119</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">c<sub>64</sub></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">c<sub>64</sub></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="E4:121_4_1_3"/><i><font color="Green">E120</font></i>: 
( <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>63</sub></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">c<sub>63</sub></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">c<sub>66</sub></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">c<sub>64</sub></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">c<sub>64</sub></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">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="toprealb.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E5:121_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">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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#K33">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">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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#K33">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">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p5"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p5"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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="ref" href="toprealb.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E6:121_4_1_3"/><b>then </b><i><font color="Green">E121</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#K33">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">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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#K33">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">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="toprealb.html#E7:121_4"><i><font color="Green">E112</font></i></a>, <a class="txt" href="toprealb.html#E4:121_4_1_3"><i><font color="Green">E120</font></i></a>, <a class="ref" href="sppol_2.html#T1">SPPOL_2:1</a></i>;<br/><a NAME="E7:121_4_1_3"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">c<sub>63</sub></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">c<sub>66</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="toprealb.html#E11:121_4"><i><font color="Green">E115</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:121_4_1_3"/><i><font color="Green">E123</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>66</sub></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">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="toprealb.html#E1:121_4_1_3"><i><font color="Green">E118</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E9:121_4_1_3"/><b>then </b>
<font color="Maroon">c<sub>66</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E10:121_4_1_3"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">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#K33">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">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E11:121_4_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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">c<sub>66</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E7:121_4_1_3"><i><font color="Green">E122</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E12:121_4_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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="E13:121_4_1_3"/><b>then </b><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">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">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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#K33">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="E14:121_4_1_3"/>
<font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>64</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E13:121_4"><i><font color="Green">E117</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E15:121_4_1_3"/><b>then </b><i><font color="Green">E126</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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="E16:121_4_1_3"/>
<font color="Maroon">c<sub>64</sub></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">c<sub>64</sub></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="E17:121_4_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>64</sub></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">c<sub>64</sub></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">c<sub>64</sub></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">c<sub>64</sub></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="E18:121_4_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>64</sub></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">c<sub>64</sub></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="E19:121_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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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#K33">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="E20:121_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#K33">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">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E6:121_4_1_3"><i><font color="Green">E121</font></i></a>, <a class="txt" href="toprealb.html#E10:121_4_1_3"><i><font color="Green">E124</font></i></a>, <a class="txt" href="toprealb.html#E13:121_4_1_3"><i><font color="Green">E125</font></i></a>, <a class="txt" href="toprealb.html#E15:121_4_1_3"><i><font color="Green">E126</font></i></a>, <a class="ref" href="complex2.html#T12">COMPLEX2:12</a></i>;<br/><a NAME="E21:121_4_1_3"/><b>then </b><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>64</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>64</sub></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="E22:121_4_1_3"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>64</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span>
 ;<br/><a NAME="E23:121_4_1_3"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>66</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E8:121_4_1_3"><i><font color="Green">E123</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E24:121_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="E25:121_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">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E23:121_4_1_3"><i><font color="Green">E129</font></i></a></i>;<br/><a NAME="E26:121_4_1_3"/><b>then </b><i><font color="Green">E130</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">c<sub>64</sub></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">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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">c<sub>64</sub></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="E27:121_4_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>63</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>66</sub></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="txt" href="toprealb.html#E12:121_4"><i><font color="Green">E116</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E28:121_4_1_3"/><b>then </b>
<font color="Maroon">c<sub>64</sub></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">c<sub>64</sub></font></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E22:121_4_1_3"><i><font color="Green">E128</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E29:121_4_1_3"/><b>then </b>
<font color="Maroon">c<sub>64</sub></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">c<sub>64</sub></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">c<sub>64</sub></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="txt" href="toprealb.html#E21:121_4_1_3"><i><font color="Green">E127</font></i></a>, <a class="txt" href="toprealb.html#E26:121_4_1_3"><i><font color="Green">E130</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E30:121_4_1_3"/>
<font color="Maroon">c<sub>64</sub></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="txt" href="toprealb.html#E3:121_4_1_3"><i><font color="Green">E119</font></i></a>, <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="E13:121"/>
 <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">c<sub>62</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E6:121"><i><font color="Green">E107</font></i></a>, <a class="txt" href="toprealb.html#E11:121"><i><font color="Green">E108</font></i></a>, <a class="ref" href="t_0topsp.html#T2">T_0TOPSP:2</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>63</sub></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="E14:121"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>63</sub></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">c<sub>64</sub></font> be   <a href="struct_0.html#NM7">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">c<sub>63</sub></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">c<sub>62</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E15:121"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>64</sub></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">c<sub>63</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>65</sub></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="E17:121"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>63</sub></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">c<sub>65</sub></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">c<sub>65</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E14:121"><i><font color="Green">E109</font></i></a></i>;<br/>
<a NAME="E18:121"/>
( <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">c<sub>65</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>65</sub></font> &amp; 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>65</sub></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">c<sub>65</sub></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">c<sub>66</sub></font> = <font color="Maroon">c<sub>63</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="toprealb.html#E17:121"><i><font color="Green">E111</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E20:121"/><i><font color="Green">E112</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">c<sub>67</sub></font> = <font color="Maroon">c<sub>62</sub></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="txt" href="toprealb.html#E11:121"><i><font color="Green">E108</font></i></a>, <a class="txt" href="toprealb.html#E20:121"><i><font color="Green">E112</font></i></a>, <a class="txt" href="toprealb.html#E62"><i><font color="Green">Lemma43</font></i></a>, <a class="txt" href="toprealb.html#E67"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>68</sub></font> = <font color="Maroon">c<sub>64</sub></font> as   <a href="struct_0.html#NM7">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">c<sub>66</sub></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">c<sub>67</sub></font></span>)</span> ;<br/>
<a NAME="E23:121"/><i><font color="Green">E113</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">c<sub>62</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>62</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E24:121"/><i><font color="Green">E114</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">c<sub>63</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E14:121"><i><font color="Green">E109</font></i></a>, <a class="ref" href="toprealb.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E25:121"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>64</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E14:121"><i><font color="Green">E109</font></i></a>, <a class="txt" href="toprealb.html#E15:121"><i><font color="Green">E110</font></i></a>, <a class="txt" href="toprealb.html#E6"><i><font color="Green">Lemma4</font></i></a>, <a class="ref" href="toprealb.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E26:121"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>64</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E15:121"><i><font color="Green">E110</font></i></a>, <a class="ref" href="topreala.html#T29">TOPREALA:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>69</sub></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">c<sub>65</sub></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">c<sub>65</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>5</sub></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">c<sub>70</sub></font> = <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>69</sub></font> as   <a href="struct_0.html#NM7">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">c<sub>69</sub></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#E73"><i><font color="Green">Lemma51</font></i></a></i>;<br/>
<a NAME="E29:121"/>
<font color="Maroon">c<sub>63</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>69</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E17:121"><i><font color="Green">E111</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>71</sub></font> = <font color="Maroon">c<sub>63</sub></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">c<sub>69</sub></font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E31:121"/><i><font color="Green">E117</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">c<sub>69</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>71</sub></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">c<sub>63</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E17:121"><i><font color="Green">E111</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<a NAME="E32:121"/><i><font color="Green">E118</font></i>: 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>63</sub></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">c<sub>69</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>66</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E17:121"><i><font color="Green">E111</font></i></a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>;<br/>
<a NAME="E33:121"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>68</sub></font> is <a href="funct_2.html#V2">onto</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:121_5"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>68</sub></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">c<sub>67</sub></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">c<sub>72</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:121_5"/>
<font color="Maroon">c<sub>72</sub></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">c<sub>67</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>73</sub></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="E4:121_5"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>73</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>63</sub></font>
 <b>and </b><br/><a NAME="E5:121_5"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>72</sub></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">c<sub>73</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E6:121"><i><font color="Green">E107</font></i></a>, <a class="txt" href="toprealb.html#E11:121"><i><font color="Green">E108</font></i></a>, <a class="txt" href="toprealb.html#E23:121"><i><font color="Green">E113</font></i></a>, <a class="txt" href="toprealb.html#E24:121"><i><font color="Green">E114</font></i></a>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E6:121_5"/><i><font color="Green">E122</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">c<sub>63</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>63</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E62"><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="E7:121_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">c<sub>63</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>73</sub></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">c<sub>73</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E4:121_5"><i><font color="Green">E120</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E8:121_5"/>
<font color="Maroon">c<sub>72</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>68</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E15:121"><i><font color="Green">E110</font></i></a>, <a class="txt" href="toprealb.html#E4:121_5"><i><font color="Green">E120</font></i></a>, <a class="txt" href="toprealb.html#E5:121_5"><i><font color="Green">E121</font></i></a>, <a class="txt" href="toprealb.html#E6:121_5"><i><font color="Green">E122</font></i></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">c<sub>72</sub></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">c<sub>65</sub></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="E35:121"/>
 <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">c<sub>65</sub></font></span>)</span></span>)</span> is <a href="t_0topsp.html#V1">open</a>
 
;<br/>
<a NAME="E36:121"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>70</sub></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="E37:121"/>
<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">c<sub>65</sub></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">c<sub>65</sub></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="E38:121"/><b>then </b>
<font color="Maroon">c<sub>64</sub></font> is <a href="t_0topsp.html#V1">open</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E15:121"><i><font color="Green">E110</font></i></a>, <a class="txt" href="toprealb.html#E31:121"><i><font color="Green">E117</font></i></a>, <a class="txt" href="toprealb.html#E32:121"><i><font color="Green">E118</font></i></a>, <a class="txt" href="toprealb.html#E33:121"><i><font color="Green">E119</font></i></a>, <a class="txt" href="toprealb.html#E36:121"><i><font color="Green">E120</font></i></a>, <a class="ref" href="topreala.html#T31">TOPREALA:31</a></i>;<br/>
<b>hence </b><a NAME="E39:121"/>
<font color="Maroon">c<sub>64</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="toprealb.html#E25:121"><i><font color="Green">E115</font></i></a>, <a class="txt" href="toprealb.html#E26:121"><i><font color="Green">E116</font></i></a>, <a class="txt" href="toprealb.html#E33:121"><i><font color="Green">E119</font></i></a>, <a class="ref" href="topreala.html#T37">TOPREALA:37</a></i>;<br/>


</div>
