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

<a NAME="E1:80_1_1"/><i><font color="Green">E61</font></i>: 
 <a href="jordan16.html#K1">AffineMap</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span>,0 <a href="hidden.html#R1">=</a>  <a href="toprealb.html#K6" target="_self">R^1</a> <span class="p1">(<span class="default"><a href="jordan16.html#K1">AffineMap</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span>,0</span>)</span>
 
;<br/>
<a NAME="E2:80_1_1"/>
 <a href="relset_1.html#K4">dom</a> <a href="sin_cos.html#K19">sin</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="sin_cos.html#D20">SIN_COS:def 20</a></i>;<br/>
<a NAME="E3:80_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="sin_cos.html#K19">sin</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topalg_2.html#K3">R^1</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:80_1_1"/>
 <a href="relset_1.html#K4">dom</a> <a href="sin_cos.html#K22">cos</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="sin_cos.html#D22">SIN_COS:def 22</a></i>;<br/>
<a NAME="E5:80_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <a href="sin_cos.html#K22">cos</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topalg_2.html#K3">R^1</a> 
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">sR</font> =  <a href="toprealb.html#K6" target="_self">R^1</a> <a href="sin_cos.html#K19">sin</a> ;<br/>
<b>set </b><font color="Maroon">cR</font> =  <a href="toprealb.html#K6" target="_self">R^1</a> <a href="sin_cos.html#K22">cos</a> ;<br/>
<b>reconsider </b><font color="Maroon">sR</font> =  <a href="toprealb.html#K6" target="_self">R^1</a> <a href="sin_cos.html#K19">sin</a> , <font color="Maroon">cR</font> =  <a href="toprealb.html#K6" target="_self">R^1</a> <a href="sin_cos.html#K22">cos</a>  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>, </i>;<br/>
<b>reconsider </b><font color="Maroon">l</font> =  <a href="jordan16.html#K1">AffineMap</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span>,0 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/>;<br/>
<a NAME="E8:80_1_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan16.html#K1">AffineMap</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span>,0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:80_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1">AffineMap</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span>,0</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topalg_2.html#K3">R^1</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E10:80_1_1"/><i><font color="Green">E71</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jordan16.html#K1">AffineMap</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span>,0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="jordan16.html#T32">JORDAN16:32</a></i>;<br/>
<b>set </b><font color="Maroon">s</font> = <font color="Maroon">sR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font>;<br/>
<b>set </b><font color="Maroon">c</font> = <font color="Maroon">cR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font>;<br/>
<b>reconsider </b><font color="Maroon">g</font> =  <a href="toprealb.html#K12" target="_self">CircleMap</a>  as   <a href="struct_0.html#NM6">Function</a> of <a href="topalg_2.html#K3">R^1</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreala.html#T28">TOPREALA:28</a></i>;<br/>
<a NAME="E12:80_1_1"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topalg_2.html#K3">R^1</a> <br/> for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> holds <br/> ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:80_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E2:80_1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E3:80_1_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E6:80_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E7:80_1_1_1"/><i><font color="Green">E77</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">e</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, , <a class="ref" href="goboard6.html#T8">GOBOARD6:8</a></i>;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<b>set </b><font color="Maroon">F</font> = 1,2 <a href="funct_4.html#K5">--&gt;</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"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<a NAME="E8:80_1_1_1"/>
 <a href="square_1.html#K7">sqrt</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="square_1.html#T84">SQUARE_1:84</a></i>;<br/>
<a NAME="E9:80_1_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="toprealb.html#E12"><i><font color="Green">Lemma59</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E10:80_1_1_1"/><i><font color="Green">E116</font></i>: 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K5">--&gt;</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">e</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topreal6.html#T49">TOPREAL6:49</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>, <font color="Maroon">B</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E12:80_1_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="jordan6.html#T46">JORDAN6:46</a></i>;<br/>
<a NAME="E13:80_1_1_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">cR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">sR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E7"><i><font color="Green">Lemma50</font></i></a></i>;<br/>
<a NAME="E14:80_1_1_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">sR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E15:80_1_1_1"/>
<font color="Maroon">sR</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E9"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/>
<a NAME="E16:80_1_1_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">sR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E8"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="toprealb.html#E11"><i><font color="Green">Lemma54</font></i></a>, , , <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E17:80_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">sR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:80_1_1"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="topreal6.html#T20">TOPREAL6:20</a></i>;<br/>
<b>then consider </b><font color="Maroon">Ws</font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E19:80_1_1_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ws</font>
 <b>and </b><br/><a NAME="E20:80_1_1_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">Ws</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E21:80_1_1_1"/><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">sR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Ws</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="toprealb.html#T2" target="_self">Th2</a>, <a class="txt" href="toprealb.html#E3:80_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<a NAME="E22:80_1_1_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="jordan6.html#T46">JORDAN6:46</a></i>;<br/>
<a NAME="E23:80_1_1_1"/><i><font color="Green">E131</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">cR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E24:80_1_1_1"/>
<font color="Maroon">cR</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E10"><i><font color="Green">Lemma53</font></i></a>, <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/>
<a NAME="E25:80_1_1_1"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">cR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E8"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="toprealb.html#E11"><i><font color="Green">Lemma54</font></i></a>, , , <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E26:80_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">cR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:80_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="toprealb.html#E10:80_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="topreal6.html#T20">TOPREAL6:20</a></i>;<br/>
<b>then consider </b><font color="Maroon">Wc</font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E28:80_1_1_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Wc</font>
 <b>and </b><br/><a NAME="E29:80_1_1_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">Wc</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E30:80_1_1_1"/><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">cR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Wc</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="txt" href="toprealb.html#E1:80_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<b>set </b><font color="Maroon">W</font> = <font color="Maroon">Ws</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Wc</font>;<br/>
<b>take </b>
<font color="Maroon">Ws</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Wc</font>
;<br/>

<b>thus </b><a NAME="E31:80_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ws</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Wc</font>
 <b>by </b><i>, <a class="txt" href="toprealb.html#E2:80_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<b>thus </b><a NAME="E32:80_1_1_1"/>
<font color="Maroon">Ws</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Wc</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="toprealb.html#E5:80_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="toprealb.html#E3:80_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>

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

<b>assume </b><a NAME="E33:80_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">Ws</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Wc</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</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="E35:80_1_1_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ws</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Wc</font>
 <b>and </b><br/><a NAME="E36:80_1_1_1"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E37:80_1_1_1"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">cR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">sR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E7"><i><font color="Green">Lemma50</font></i></a></i>;<br/>
<a NAME="E38:80_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ws</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E39:80_1_1_1"/><b>then </b><i><font color="Green">E181</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">sR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">sR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Ws</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E40:80_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Wc</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E41:80_1_1_1"/><b>then </b><i><font color="Green">E182</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">cR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">cR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Wc</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E42:80_1_1_1"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">cR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">sR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> 1,2 <a href="funct_4.html#K5">--&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">cR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">sR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreala.html#T49">TOPREALA:49</a></i>;<br/>
<a NAME="E43:80_1_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">cR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">sR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K5">--&gt;</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E9:80_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="toprealb.html#E6:80_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="toprealb.html#T5" target="_self">Th5</a>, <a class="ref" href="toprealb.html#T6" target="_self">Th6</a>, <a class="ref" href="hilbert3.html#T11">HILBERT3:11</a></i>;<br/>
<a NAME="E44:80_1_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">cR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">sR</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">e</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="toprealb.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E45:80_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="txt" href="toprealb.html#E7:80_1_1_1"><i><font color="Green">E77</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:80_1_1"/><b>then </b>
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<b>hence </b><a NAME="E14:80_1_1"/>
 <a href="toprealb.html#K12" target="_self">CircleMap</a>  is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T8">TOPMETR:8</a></i>;<br/>


</div>
