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

<b>set </b><font color="Maroon">f</font> =  <a href="topalg_5.html#K7" target="_self">Ciso</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="gr_cy_1.html#K3">INT.Group</a> , <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="gr_cy_1.html#K3">INT.Group</a> ;<br/><b>consider </b><font color="Maroon">fX</font> being   <a href="topalg_1.html#NM1">Loop</a> of  <a href="toprealb.html#K9">c[10]</a> , <font color="Maroon">fY</font> being   <a href="topalg_1.html#NM1">Loop</a> of  <a href="toprealb.html#K9">c[10]</a> <b> such that </b><br/><a NAME="E2:45_1_1_1"/><i><font color="Green">E72</font></i>: 
<a href="topalg_5.html#K7" target="_self">Ciso</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9">c[10]</a> </span>)</span>,<font color="Maroon">fX</font>
 <b>and </b><br/><a NAME="E3:45_1_1_1"/><i><font color="Green">E73</font></i>: 
<a href="topalg_5.html#K7" target="_self">Ciso</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9">c[10]</a> </span>)</span>,<font color="Maroon">fY</font>
 <b>and </b><br/><a NAME="E4:45_1_1_1"/><i><font color="Green">E75</font></i>: 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9">c[10]</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K7" target="_self">Ciso</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="topalg_5.html#K7" target="_self">Ciso</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9">c[10]</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">fX</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">fY</font></span>)</span>
 <b>by </b><i><a class="ref" href="topalg_1.html#D3">TOPALG_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">tx</font> =  <a href="jordan16.html#K1">AffineMap</a> 1,<font color="Maroon">x</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topalg_2.html#K3">R^1</a> ,<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>set </b><font color="Maroon">p</font> = <font color="Maroon">tx</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">y</font></span>)</span>;<br/><a NAME="E6:45_1_1_1"/>
<font color="Maroon">tx</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">y</font></span>)</span> is    <a href="borsuk_2.html#M1">Path</a> of  <a href="toprealb.html#K4">R^1</a> <font color="Maroon">x</font>, <a href="toprealb.html#K4">R^1</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:45_1_1_1_1"/>
<font color="Maroon">x</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E2:45_1_1_1_1"/><b>then </b>
<font color="Maroon">tx</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="ref" href="jordan16.html#T35">JORDAN16:35</a></i>;<br/>
<a NAME="E3:45_1_1_1_1"/><b>then </b>
<font color="Maroon">tx</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:45_1_1_1_1"/>
<font color="Maroon">tx</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">y</font></span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D4">BORSUK_2:def 4</a><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">tx</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0 = 
<font color="Maroon">tx</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">j0</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">tx</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j0</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="jordan16.html#D3">JORDAN16:def 3</a></i>
<br/>.= 
 <a href="toprealb.html#K4">R^1</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="toprealb.html#D2">TOPREALB:def 2</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">tx</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 1 = 
<font color="Maroon">tx</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">j1</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">tx</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="jordan16.html#D3">JORDAN16:def 3</a></i>
<br/>.= 
 <a href="toprealb.html#K4">R^1</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="toprealb.html#D2">TOPREALB:def 2</a></i>
;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">tx</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">y</font></span>)</span> as    <a href="borsuk_2.html#M1">Path</a> of  <a href="toprealb.html#K4">R^1</a> <font color="Maroon">x</font>, <a href="toprealb.html#K4">R^1</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span> ;<br/><a NAME="E8:45_1_1_1"/>
<a href="topalg_5.html#K7" target="_self">Ciso</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9">c[10]</a> </span>)</span>,<span class="p1">(<span class="default"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E9:45_1_1_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">fX</font>, <a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">x</font> <a href="borsuk_2.html#R4">are_homotopic</a> 
 <b>by </b><i>, <a class="ref" href="topalg_1.html#T47">TOPALG_1:47</a></i>;<br/><a NAME="E10:45_1_1_1"/>
<a href="topalg_5.html#K7" target="_self">Ciso</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9">c[10]</a> </span>)</span>,<span class="p1">(<span class="default"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E11:45_1_1_1"/><b>then </b>
<font color="Maroon">fY</font>, <a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">y</font> <a href="borsuk_2.html#R4">are_homotopic</a> 
 <b>by </b><i>, <a class="ref" href="topalg_1.html#T47">TOPALG_1:47</a></i>;<br/><a NAME="E12:45_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">fX</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">fY</font>,<span class="p1">(<span class="default"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">x</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">y</font></span>)</span> <a href="borsuk_2.html#R4">are_homotopic</a> 
 <b>by </b><i>, <a class="ref" href="borsuk_6.html#T84">BORSUK_6:84</a></i>;<br/><a NAME="E13:45_1_1_1"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">a</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds  <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">x</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">x</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_1_1_2_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 or 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:45_1_1_1_2_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:45_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font> is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T6">BORSUK_6:6</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">x</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> = 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">x</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#D4">BORSUK_6:def 4</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K20">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</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">TOPREALB:def 11</a></i>
<br/>.= 
<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"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</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"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">x</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#D4">BORSUK_6:def 4</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:45_1_1_1_2_1_2"/><i><font color="Green">E84</font></i>: 
1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:45_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T7">BORSUK_6:7</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">x</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> = 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">x</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#D4">BORSUK_6:def 4</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">tx</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">y</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">tx</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan16.html#D3">JORDAN16:def 3</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K20">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></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">TOPREALB:def 11</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</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"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="complex2.html#T10">COMPLEX2:10</a></i>
<br/>.= 
<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"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</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"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="complex2.html#T9">COMPLEX2:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">x</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#D4">BORSUK_6:def 4</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a href="topalg_5.html#K7" target="_self">Ciso</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> = 
<a href="topalg_5.html#K7" target="_self">Ciso</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T14">GR_CY_1:14</a></i>
<br/>.= 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9">c[10]</a> </span>)</span>,<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topalg_5.html#K2" target="_self">ExtendInt</a> <font color="Maroon">x</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9">c[10]</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">x</font></span>)</span> <a href="borsuk_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K6" target="_self">cLoop</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_5.html#K7" target="_self">Ciso</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K7" target="_self">Ciso</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="topalg_1.html#T47">TOPALG_1:47</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:45_1_1"/>
 <a href="topalg_5.html#K7" target="_self">Ciso</a>  is   <a href="group_6.html#NM2">Homomorphism</a> of <a href="gr_cy_1.html#K3">INT.Group</a> ,<span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>,<a href="toprealb.html#K9">c[10]</a> </span>)</span>
 <b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/>


</div>
