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

<b>set </b><font color="Maroon">c<sub>41</sub></font> = <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"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p3">(<span class="default">0 <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></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>42</sub></font> = <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default">0,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_2.html#K2">.]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>43</sub></font> = <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,0</span><a href="rcomp_2.html#K1">.[</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>44</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>45</sub></font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> ;<br/>
<a NAME="E1:106"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K9" target="_self">c[10]</a> </span>)</span> holds  <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>46</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K9" target="_self">c[10]</a> </span>)</span>;<br/>

<b>consider </b><font color="Maroon">c<sub>47</sub></font>, <font color="Maroon">c<sub>48</sub></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="E2:106_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>47</sub></font>,<font color="Maroon">c<sub>48</sub></font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E3:106_1"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 implies <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K6">arccos</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> <a href="real_1.html#K6">/</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> )
 <b>and </b><br/><a NAME="E4:106_1"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 implies <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6">arccos</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="toprealb.html#D13" target="_self">Def13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>49</sub></font> = <font color="Maroon">c<sub>46</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E11"><i><font color="Green">Lemma9</font></i></a></i>;<br/>
<a NAME="E6:106_1"/>
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>49</sub></font> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="toprealb.html#E2:106_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E7:106_1"/><b>then </b>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>47</sub></font> &amp; <font color="Maroon">c<sub>47</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="toprealb.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E8:106_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E9:106_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>47</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<i><font color="Green">E91</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> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p4">(<span class="default">0 <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></span>)</span></span>)</span> = 
 <a href="toprealb.html#K5" target="_self">R^1</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default">0 <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>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default">0 <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>

;<br/>
<a NAME="E11:106_1"/><i><font color="Green">E92</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <a href="sin_cos6.html#K4">arccos</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>
<b>then </b><i><font color="Green">E93</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>47</sub></font> = 
<span class="p1">(<span class="default"><a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E8:106_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a>, <a class="ref" href="sin_cos6.html#T88">SIN_COS6:88</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>47</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</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>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
;<br/>
<a NAME="E13:106_1"/><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a>  <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="fcont_1.html#T21">FCONT_1:21</a>, <a class="ref" href="sin_cos6.html#T109">SIN_COS6:109</a></i>;<br/>
<a NAME="E14:106_1"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Maroon">c<sub>47</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E8:106_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="fcont_1.html#D2">FCONT_1:def 2</a></i>;<br/>
<a NAME="E15:106_1"/>
 <a href="toprealb.html#K7" target="_self">Tcircle</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span>,1 is    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="topreala.html#K1">Trectangle</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="toprealb.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E16:106_1"/><b>then </b>
 <a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2 is    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="topreala.html#K1">Trectangle</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<a NAME="E17:106_1"/><b>then </b><i><font color="Green">E96</font></i>: 
 <a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K9" target="_self">c[10]</a>  is    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="topreala.html#K1">Trectangle</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T7">TSEP_1:7</a></i>;<br/>
<a NAME="E18:106_1"/><i><font color="Green">E97</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E4"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E19:106_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>49</sub></font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="toprealb.html#E2:106_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:106_1_3"/>
( <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or <font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_3_1"><b>suppose </b></a><a NAME="E1:106_1_3_1"/>
<font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:106_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a>  <a href="toprealb.html#K10" target="_self">c[-10]</a> 
 <b>by </b><i><a class="txt" href="toprealb.html#E19:106_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="toprealb.html#T25" target="_self">Th25</a></i>;<br/><b>hence </b><a NAME="E3:106_1_3_1"/>
 <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">c<sub>46</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E126"><i><font color="Green">Lemma83</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_3_2"><b>suppose </b></a><a NAME="E1:106_1_3_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:106_1_3_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <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">0,<span class="p4">(<span class="default">0 <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></span>)</span></span>)</span> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K9" target="_self">c[10]</a> </span>)</span> st <br/>( <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>50</sub></font> be   <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">0,<span class="p4">(<span class="default">0 <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></span>)</span></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:106_1_3_2_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>50</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:106_1_3_2_1"/><i><font color="Green">E101</font></i>: 
<a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>50</sub></font>
 ;<br/>

<a NAME="E3:106_1_3_2_1"/><i><font color="Green">E102</font></i>: 
<a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E4:106_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="toprealb.html#E1:106_1_3_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="sin_cos6.html#D4">SIN_COS6:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>51</sub></font> = <font color="Maroon">c<sub>50</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="toprealb.html#E10:106_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>52</sub></font> = <font color="Maroon">c<sub>51</sub></font> 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="E6:106_1_3_2_1"/>
<font color="Maroon">c<sub>52</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:106_1_3_2_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="tsep_1.html#T17">TSEP_1:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>53</sub></font> = <font color="Maroon">c<sub>51</sub></font> as  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="borsuk_5.html#T62">BORSUK_5:62</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>54</sub></font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>46</sub></font><b> such that </b><br/><a NAME="E9:106_1_3_2_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>54</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>53</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E2:106_1_3_2_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="rcomp_1.html#T39">RCOMP_1:39</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>55</sub></font> = <font color="Maroon">c<sub>40</sub></font> <a href="seq_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span>;<br/>
<a NAME="E10:106_1_3_2_1"/><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="seq_1.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>47</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E8:106_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<i><font color="Green">E105</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>40</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
<a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="toprealb.html#E11:106_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="sin_cos6.html#T88">SIN_COS6:88</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>then </b><i><font color="Green">E106</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>47</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>47</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E8:106_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>47</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E8:106_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="toprealb.html#E12:106_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E13:106_1_3_2_1"/>
<font color="Maroon">c<sub>40</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="topreala.html#T38">TOPREALA:38</a></i>;<br/>
<a NAME="E14:106_1_3_2_1"/><b>then </b>
<font color="Maroon">c<sub>40</sub></font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/>
<a NAME="E15:106_1_3_2_1"/><b>then </b>
<font color="Maroon">c<sub>40</sub></font> <a href="seq_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E13:106_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="fcont_1.html#T19">FCONT_1:19</a></i>;<br/>
<a NAME="E16:106_1_3_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Maroon">c<sub>47</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E8:106_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="fcont_1.html#D2">FCONT_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>56</sub></font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>47</sub></font><b> such that </b><br/><a NAME="E18:106_1_3_2_1"/><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="seq_1.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>56</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>54</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E3:106_1_3_2_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="toprealb.html#E10:106_1_3_2_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="toprealb.html#E12:106_1_3_2_1"><i><font color="Green">E106</font></i></a>, <a class="ref" href="fcont_1.html#T5">FCONT_1:5</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>57</sub></font> = <font color="Maroon">c<sub>56</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>;<br/>
<a NAME="E19:106_1_3_2_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>56</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>58</sub></font> = <font color="Maroon">c<sub>56</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>, <font color="Maroon">c<sub>59</sub></font> = <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,0</span><a href="rcomp_2.html#K1">.[</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E4"><i><font color="Green">Lemma2</font></i></a>, <a class="ref" href="topreala.html#T11">TOPREALA:11</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>60</sub></font> = <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default">1,2 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>58</sub></font>,<font color="Maroon">c<sub>59</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K9" target="_self">c[10]</a> </span>)</span>;<br/>
<a NAME="E21:106_1_3_2_1"/>
<font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i><a class="txt" href="toprealb.html#E19:106_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="toprealb.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E22:106_1_3_2_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>59</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:106_1_3_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<a NAME="E23:106_1_3_2_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> 1,2 <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c<sub>47</sub></font>,<font color="Maroon">c<sub>48</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E2:106_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="topreala.html#T49">TOPREALA:49</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>61</sub></font> = <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default">1,2 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>58</sub></font>,<font color="Maroon">c<sub>59</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K9" target="_self">c[10]</a> </span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K9" target="_self">c[10]</a> </span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>61</sub></font>
;<br/>

<a NAME="E25:106_1_3_2_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>58</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>56</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>62</sub></font> =  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>58</sub></font>,<font color="Maroon">c<sub>59</sub></font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topreala.html#K1">Trectangle</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topreala.html#T60">TOPREALA:60</a></i>;<br/>
<a NAME="E27:106_1_3_2_1"/>
 <a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">c<sub>56</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>56</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>63</sub></font> = <font color="Maroon">c<sub>58</sub></font> as  <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="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E18:106_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>64</sub></font> = <font color="Maroon">c<sub>59</sub></font> as  <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="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="ref" href="topreala.html#T47">TOPREALA:47</a></i>;<br/>
<a NAME="E30:106_1_3_2_1"/>
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R1">=</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> <font color="Maroon">c<sub>63</sub></font>,<font color="Maroon">c<sub>64</sub></font></span>)</span>
 
;<br/>
<a NAME="E31:106_1_3_2_1"/><b>then </b>
<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="topreala.html#T61">TOPREALA:61</a></i>;<br/>
<b>hence </b><a NAME="E32:106_1_3_2_1"/>
<font color="Maroon">c<sub>61</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="toprealb.html#E17:106_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="toprealb.html#E54"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>

<a NAME="E33:106_1_3_2_1"/>
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>56</sub></font>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T37">RCOMP_1:37</a></i>;<br/>
<a NAME="E34:106_1_3_2_1"/><b>then </b>
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>58</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E8:106_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E35:106_1_3_2_1"/><b>then </b>
<font color="Maroon">c<sub>46</sub></font> <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> <font color="Maroon">c<sub>58</sub></font>,<font color="Maroon">c<sub>59</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E22:106_1_3_2_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="toprealb.html#E23:106_1_3_2_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="hilbert3.html#T11">HILBERT3:11</a></i>;<br/>
<b>hence </b><a NAME="E36:106_1_3_2_1"/>
<font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>61</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>65</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="E37:106_1_3_2_1"/>
<font color="Maroon">c<sub>65</sub></font> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>61</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>66</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K9" target="_self">c[10]</a> </span>)</span><b> such that </b><br/><a NAME="E39:106_1_3_2_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>66</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>61</sub></font>
 <b>and </b><br/><a NAME="E40:106_1_3_2_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>65</sub></font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>66</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>67</sub></font>, <font color="Maroon">c<sub>68</sub></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="E42:106_1_3_2_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>66</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>67</sub></font>,<font color="Maroon">c<sub>68</sub></font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E43:106_1_3_2_1"/>
( <font color="Maroon">c<sub>68</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 implies <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>66</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K6">arccos</a> <font color="Maroon">c<sub>67</sub></font></span>)</span> <a href="real_1.html#K6">/</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> )
 <b>and </b><br/><a NAME="E44:106_1_3_2_1"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">c<sub>68</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 implies <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>66</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6">arccos</a> <font color="Maroon">c<sub>67</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="toprealb.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E45:106_1_3_2_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>66</sub></font> <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> <font color="Maroon">c<sub>58</sub></font>,<font color="Maroon">c<sub>59</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E39:106_1_3_2_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E46:106_1_3_2_1"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">c<sub>66</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>58</sub></font>
 
<b>by </b><i><a class="ref" href="topreala.html#T24">TOPREALA:24</a></i>;<br/>
<a NAME="E47:106_1_3_2_1"/><i><font color="Green">E118</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="seq_1.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E11:106_1_3_2_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E48:106_1_3_2_1"/><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="seq_1.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>58</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="seq_1.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>56</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E25:106_1_3_2_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<i><font color="Green">E120</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="seq_1.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>66</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="seq_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>66</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E19:106_1_3_2_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="toprealb.html#E46:106_1_3_2_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>66</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>66</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E11:106_1_3_2_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="toprealb.html#E19:106_1_3_2_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="toprealb.html#E46:106_1_3_2_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="seq_1.html#D4">SEQ_1:def 4</a></i>
<br/>.= 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>66</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E46:106_1_3_2_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>66</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E11:106_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="toprealb.html#E19:106_1_3_2_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="toprealb.html#E46:106_1_3_2_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a>, <a class="ref" href="sin_cos6.html#T88">SIN_COS6:88</a></i>
<br/>.= 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>67</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E42:106_1_3_2_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="topreala.html#T50">TOPREALA:50</a></i>
<br/>.= 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6">arccos</a> <font color="Maroon">c<sub>67</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos6.html#D4">SIN_COS6:def 4</a></i>
;<br/>
<a NAME="E50:106_1_3_2_1"/>
<font color="Maroon">c<sub>66</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>59</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E45:106_1_3_2_1"><i><font color="Green">E116</font></i></a>, <a class="ref" href="topreala.html#T24">TOPREALA:24</a></i>;<br/>
<a NAME="E51:106_1_3_2_1"/><b>then </b>
<font color="Maroon">c<sub>68</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>59</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E42:106_1_3_2_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="topreala.html#T50">TOPREALA:50</a></i>;<br/>
<a NAME="E52:106_1_3_2_1"/><b>then </b>
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K6">arccos</a> <font color="Maroon">c<sub>67</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>65</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E40:106_1_3_2_1"><i><font color="Green">E113</font></i></a>, <a class="txt" href="toprealb.html#E44:106_1_3_2_1"><i><font color="Green">E115</font></i></a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a>, <a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/>
<a NAME="E53:106_1_3_2_1"/><b>then </b>
<font color="Maroon">c<sub>65</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="seq_1.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>58</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E19:106_1_3_2_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="toprealb.html#E46:106_1_3_2_1"><i><font color="Green">E117</font></i></a>, <a class="txt" href="toprealb.html#E47:106_1_3_2_1"><i><font color="Green">E118</font></i></a>, <a class="txt" href="toprealb.html#E49:106_1_3_2_1"><i><font color="Green">E120</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E54:106_1_3_2_1"/><b>then </b>
<font color="Maroon">c<sub>65</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="seq_1.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>56</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E48:106_1_3_2_1"><i><font color="Green">E119</font></i></a></i>;<br/>
<a NAME="E55:106_1_3_2_1"/><b>then </b>
<font color="Maroon">c<sub>65</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>54</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E18:106_1_3_2_1"><i><font color="Green">E107</font></i></a></i>;<br/>
<b>hence </b><a NAME="E56:106_1_3_2_1"/>
<font color="Maroon">c<sub>65</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>50</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E9:106_1_3_2_1"><i><font color="Green">E103</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:106_1_3_2"/>
 <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">c<sub>46</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T48">TMAP_1:48</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_3_3"><b>suppose </b></a><a NAME="E1:106_1_3_3"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:106_1_3_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <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">0,<span class="p4">(<span class="default">0 <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></span>)</span></span>)</span> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K9" target="_self">c[10]</a> </span>)</span> st <br/>( <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_3_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>50</sub></font> be   <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">0,<span class="p4">(<span class="default">0 <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></span>)</span></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:106_1_3_3_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>50</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:106_1_3_3_1"/><i><font color="Green">E101</font></i>: 
<a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>50</sub></font>
 ;<br/>

<a NAME="E3:106_1_3_3_1"/><i><font color="Green">E102</font></i>: 
<a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> <a href="real_1.html#K6">/</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>
 
<b>by </b><i><a class="txt" href="toprealb.html#E3:106_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="toprealb.html#E1:106_1_3_3"><i><font color="Green">E99</font></i></a>, <a class="ref" href="sin_cos6.html#D4">SIN_COS6:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>51</sub></font> = <font color="Maroon">c<sub>50</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="toprealb.html#E10:106_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>52</sub></font> = <font color="Maroon">c<sub>51</sub></font> 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="E6:106_1_3_3_1"/>
<font color="Maroon">c<sub>52</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:106_1_3_3_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="tsep_1.html#T17">TSEP_1:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>53</sub></font> = <font color="Maroon">c<sub>51</sub></font> as  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="borsuk_5.html#T62">BORSUK_5:62</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>54</sub></font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>46</sub></font><b> such that </b><br/><a NAME="E9:106_1_3_3_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>54</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>53</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E2:106_1_3_3_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="rcomp_1.html#T39">RCOMP_1:39</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>55</sub></font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>47</sub></font><b> such that </b><br/><a NAME="E11:106_1_3_3_1"/><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>55</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>54</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E9:106_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="toprealb.html#E12:106_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="toprealb.html#E14:106_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="toprealb.html#E3:106_1_3_3_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="fcont_1.html#T5">FCONT_1:5</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>56</sub></font> = <font color="Maroon">c<sub>55</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>;<br/>
<a NAME="E12:106_1_3_3_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>55</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>57</sub></font> = <font color="Maroon">c<sub>55</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>, <font color="Maroon">c<sub>58</sub></font> = <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default">0,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_2.html#K2">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E4"><i><font color="Green">Lemma2</font></i></a>, <a class="ref" href="topreala.html#T15">TOPREALA:15</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>59</sub></font> = <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default">1,2 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>57</sub></font>,<font color="Maroon">c<sub>58</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K9" target="_self">c[10]</a> </span>)</span>;<br/>
<a NAME="E14:106_1_3_3_1"/>
<font color="Maroon">c<sub>48</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="toprealb.html#E19:106_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="toprealb.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E15:106_1_3_3_1"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>58</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:106_1_3_3"><i><font color="Green">E99</font></i></a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<a NAME="E16:106_1_3_3_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> 1,2 <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c<sub>47</sub></font>,<font color="Maroon">c<sub>48</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E2:106_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="topreala.html#T49">TOPREALA:49</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>60</sub></font> = <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default">1,2 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>57</sub></font>,<font color="Maroon">c<sub>58</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K9" target="_self">c[10]</a> </span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K9" target="_self">c[10]</a> </span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>60</sub></font>
;<br/>

<a NAME="E18:106_1_3_3_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>57</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>55</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>61</sub></font> =  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>57</sub></font>,<font color="Maroon">c<sub>58</sub></font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topreala.html#K1">Trectangle</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topreala.html#T60">TOPREALA:60</a></i>;<br/>
<a NAME="E20:106_1_3_3_1"/>
 <a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">c<sub>55</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>55</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>62</sub></font> = <font color="Maroon">c<sub>57</sub></font> as  <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="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E18:106_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>63</sub></font> = <font color="Maroon">c<sub>58</sub></font> as  <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="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span>)</span> <b>by </b><i><a class="ref" href="topreala.html#T46">TOPREALA:46</a></i>;<br/>
<a NAME="E23:106_1_3_3_1"/>
<font color="Maroon">c<sub>61</sub></font> <a href="hidden.html#R1">=</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> <font color="Maroon">c<sub>62</sub></font>,<font color="Maroon">c<sub>63</sub></font></span>)</span>
 
;<br/>
<a NAME="E24:106_1_3_3_1"/><b>then </b>
<font color="Maroon">c<sub>61</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="topreala.html#T61">TOPREALA:61</a></i>;<br/>
<b>hence </b><a NAME="E25:106_1_3_3_1"/>
<font color="Maroon">c<sub>60</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="toprealb.html#E17:106_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="toprealb.html#E54"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>

<a NAME="E26:106_1_3_3_1"/>
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>55</sub></font>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T37">RCOMP_1:37</a></i>;<br/>
<a NAME="E27:106_1_3_3_1"/><b>then </b>
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>57</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E8:106_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E28:106_1_3_3_1"/><b>then </b>
<font color="Maroon">c<sub>46</sub></font> <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> <font color="Maroon">c<sub>57</sub></font>,<font color="Maroon">c<sub>58</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E15:106_1_3_3_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="toprealb.html#E16:106_1_3_3_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="hilbert3.html#T11">HILBERT3:11</a></i>;<br/>
<b>hence </b><a NAME="E29:106_1_3_3_1"/>
<font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>60</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>64</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="E30:106_1_3_3_1"/>
<font color="Maroon">c<sub>64</sub></font> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>60</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>65</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K9" target="_self">c[10]</a> </span>)</span><b> such that </b><br/><a NAME="E32:106_1_3_3_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>65</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>60</sub></font>
 <b>and </b><br/><a NAME="E33:106_1_3_3_1"/><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#K14" target="_self">Circle2IntervalR</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>65</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>66</sub></font>, <font color="Maroon">c<sub>67</sub></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="E35:106_1_3_3_1"/><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="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>66</sub></font>,<font color="Maroon">c<sub>67</sub></font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E36:106_1_3_3_1"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">c<sub>67</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 implies <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>65</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K6">arccos</a> <font color="Maroon">c<sub>66</sub></font></span>)</span> <a href="real_1.html#K6">/</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> )
 <b>and </b><br/><a NAME="E37:106_1_3_3_1"/>
( <font color="Maroon">c<sub>67</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 implies <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>65</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6">arccos</a> <font color="Maroon">c<sub>66</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="toprealb.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E38:106_1_3_3_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>65</sub></font> <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> <font color="Maroon">c<sub>57</sub></font>,<font color="Maroon">c<sub>58</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E32:106_1_3_3_1"><i><font color="Green">E109</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E39:106_1_3_3_1"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>65</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>57</sub></font>
 
<b>by </b><i><a class="ref" href="topreala.html#T24">TOPREALA:24</a></i>;<br/>
<a NAME="E40:106_1_3_3_1"/><i><font color="Green">E115</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E11:106_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a>, <a class="ref" href="sin_cos6.html#T88">SIN_COS6:88</a></i>;<br/>
<a NAME="E41:106_1_3_3_1"/><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>57</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>55</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E18:106_1_3_3_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<i><font color="Green">E117</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>65</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>65</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E12:106_1_3_3_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="toprealb.html#E39:106_1_3_3_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>65</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E11:106_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="toprealb.html#E12:106_1_3_3_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="toprealb.html#E39:106_1_3_3_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a>, <a class="ref" href="sin_cos6.html#T88">SIN_COS6:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>66</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E35:106_1_3_3_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="topreala.html#T50">TOPREALA:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos6.html#K6">arccos</a> <font color="Maroon">c<sub>66</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos6.html#D4">SIN_COS6:def 4</a></i>
;<br/>
<a NAME="E43:106_1_3_3_1"/>
<font color="Maroon">c<sub>65</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>58</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E38:106_1_3_3_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="topreala.html#T24">TOPREALA:24</a></i>;<br/>
<a NAME="E44:106_1_3_3_1"/><b>then </b>
<font color="Maroon">c<sub>67</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>58</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E35:106_1_3_3_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="topreala.html#T50">TOPREALA:50</a></i>;<br/>
<a NAME="E45:106_1_3_3_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos6.html#K6">arccos</a> <font color="Maroon">c<sub>66</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>64</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E33:106_1_3_3_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="toprealb.html#E36:106_1_3_3_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a>, <a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/>
<a NAME="E46:106_1_3_3_1"/><b>then </b>
<font color="Maroon">c<sub>64</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>57</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E12:106_1_3_3_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="toprealb.html#E39:106_1_3_3_1"><i><font color="Green">E114</font></i></a>, <a class="txt" href="toprealb.html#E40:106_1_3_3_1"><i><font color="Green">E115</font></i></a>, <a class="txt" href="toprealb.html#E42:106_1_3_3_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E47:106_1_3_3_1"/><b>then </b>
<font color="Maroon">c<sub>64</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos6.html#K4">arccos</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>55</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E41:106_1_3_3_1"><i><font color="Green">E116</font></i></a></i>;<br/>
<a NAME="E48:106_1_3_3_1"/><b>then </b>
<font color="Maroon">c<sub>64</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>54</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E11:106_1_3_3_1"><i><font color="Green">E104</font></i></a></i>;<br/>
<b>hence </b><a NAME="E49:106_1_3_3_1"/>
<font color="Maroon">c<sub>64</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>50</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E9:106_1_3_3_1"><i><font color="Green">E103</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:106_1_3_3"/>
 <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">c<sub>46</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T48">TMAP_1:48</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:106"/>
 <a href="toprealb.html#K14" target="_self">Circle2IntervalR</a>  is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49">TMAP_1:49</a></i>;<br/>


</div>
