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

<b>let </b><font color="Maroon">c<sub>56</sub></font> be   <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#K10" target="_self">c[-10]</a> </span>)</span>;<br/>

<b>assume </b><a NAME="E1:112"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>56</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K10" target="_self">c[-10]</a> </span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  ) </span>}</span> 
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>57</sub></font> =  <a href="toprealb.html#K9" target="_self">c[10]</a>  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> ;<br/>
<a NAME="E3:112"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>57</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E4:112"/>
 <a href="toprealb.html#K9" target="_self">c[10]</a>  is   <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#K10" target="_self">c[-10]</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="toprealb.html#T21" target="_self">Th21</a>, <a class="ref" href="toprealb.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E5:112"/><b>then </b>
 <a href="toprealb.html#K9" target="_self">c[10]</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>56</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:112"><i><font color="Green">E99</font></i></a>, <a class="txt" href="toprealb.html#E3:112"><i><font color="Green">E100</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>58</sub></font> = <font color="Maroon">c<sub>56</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K10" target="_self">c[-10]</a> </span>)</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>59</sub></font> = <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K10" target="_self">c[-10]</a> </span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>58</sub></font>;<br/>
<a NAME="E7:112"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>58</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K10" target="_self">c[-10]</a> </span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>58</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E8:112"/><i><font color="Green">E102</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K10" target="_self">c[-10]</a> </span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>58</sub></font></span>)</span> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E12"><i><font color="Green">Lemma10</font></i></a></i>;<br/>
<a NAME="E9:112"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>60</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K10" target="_self">c[-10]</a> </span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>58</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E8:112"><i><font color="Green">E102</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:112"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>60</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>58</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E7:112"><i><font color="Green">E101</font></i></a></i>;<br/>
<a NAME="E12:112"/>
<span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K10" target="_self">c[-10]</a> </span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>58</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2
 
<b>by </b><i><a class="ref" href="tsep_1.html#T7">TSEP_1:7</a></i>;<br/>
<a NAME="E13:112"/><b>then </b>
<span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K10" target="_self">c[-10]</a> </span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>58</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="euclid.html#K15">TOP-REAL</a> 2
 
<b>by </b><i><a class="ref" href="tsep_1.html#T7">TSEP_1:7</a></i>;<br/>
<a NAME="E14:112"/><b>then </b><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>60</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K10" target="_self">c[-10]</a> </span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>58</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>61</sub></font> = <font color="Maroon">c<sub>38</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>60</sub></font>;<br/>
<a NAME="E15:112"/><i><font color="Green">E105</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>60</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>60</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E16:112"/><i><font color="Green">E106</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>38</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>60</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>60</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E122"><i><font color="Green">Lemma79</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E17:112"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>38</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>60</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>44</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:112_1"/>
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>38</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>60</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>63</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:112_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>63</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>38</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>60</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:112_1"/><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>38</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>60</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>63</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>62</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>64</sub></font> being   <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> such that </b><br/><a NAME="E6:112_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>64</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>63</sub></font>
 <b>and </b><br/><a NAME="E7:112_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>64</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K10" target="_self">c[-10]</a> </span>)</span>
 <b>and </b><br/><a NAME="E8:112_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>64</sub></font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="toprealb.html#E1:112"><i><font color="Green">E99</font></i></a>, <a class="txt" href="toprealb.html#E11:112"><i><font color="Green">E103</font></i></a>, <a class="txt" href="toprealb.html#E16:112"><i><font color="Green">E106</font></i></a>, <a class="txt" href="toprealb.html#E3:112_1"><i><font color="Green">E107</font></i></a></i>;<br/>
<i><font color="Green">E111</font></i>: <font color="Maroon">c<sub>62</sub></font> = 
<font color="Maroon">c<sub>38</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>63</sub></font>
<b>by </b><i><a class="txt" href="toprealb.html#E16:112"><i><font color="Green">E106</font></i></a>, <a class="txt" href="toprealb.html#E3:112_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="toprealb.html#E4:112_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>64</sub></font> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="toprealb.html#E6:112_1"><i><font color="Green">E109</font></i></a>, <a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
;<br/>
<a NAME="E10:112_1"/>
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>64</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>64</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="toprealb.html#E7:112_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="toprealb.html#T29" target="_self">Th29</a></i>;<br/>
<b>hence </b><a NAME="E11:112_1"/>
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>44</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E9:112_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>62</sub></font> = <font color="Maroon">c<sub>38</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>60</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>60</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">c<sub>44</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E15:112"><i><font color="Green">E105</font></i></a>, <a class="txt" href="toprealb.html#E16:112"><i><font color="Green">E106</font></i></a>, <a class="txt" href="toprealb.html#E132"><i><font color="Green">Lemma86</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E19:112"/>
<font color="Maroon">c<sub>62</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E123"><i><font color="Green">Lemma80</font></i></a>, <a class="ref" href="topreala.html#T29">TOPREALA:29</a></i>;<br/>
<a NAME="E20:112"/><b>then </b>
<font color="Maroon">c<sub>53</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>62</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E162"><i><font color="Green">Lemma97</font></i></a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E21:112"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>51</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>53</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>62</sub></font></span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E22:112"/>
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"><span class="p2">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K10" target="_self">c[-10]</a> </span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>58</sub></font></span>)</span> holds <span class="p1">(<span class="default"><a href="toprealb.html#K15" target="_self">Circle2IntervalL</a>  <a href="tmap_1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K10" target="_self">c[-10]</a> </span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>58</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>53</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>62</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>63</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K10" target="_self">c[-10]</a> </span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>58</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>64</sub></font> = <font color="Maroon">c<sub>63</sub></font> as   <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#K10" target="_self">c[-10]</a> </span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T2">TOPMETR:2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>65</sub></font>, <font color="Maroon">c<sub>66</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="E3:112_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>64</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>65</sub></font>,<font color="Maroon">c<sub>66</sub></font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E4:112_2"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c<sub>66</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 implies <a href="toprealb.html#K15" target="_self">Circle2IntervalL</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>64</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K3">+</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>65</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>and </b><br/><a NAME="E5:112_2"/>
( <font color="Maroon">c<sub>66</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 implies <a href="toprealb.html#K15" target="_self">Circle2IntervalL</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>64</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>65</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#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E6:112_2"/>
<font color="Maroon">c<sub>63</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#E7:112"><i><font color="Green">E101</font></i></a></i>;<br/>
<a NAME="E7:112_2"/><b>then </b><i><font color="Green">E110</font></i>: 
ex <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="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon">c<sub>63</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K10" target="_self">c[-10]</a> </span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:112"><i><font color="Green">E99</font></i></a></i>;<br/>
<a NAME="E8:112_2"/><i><font color="Green">E111</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>65</sub></font>,<font color="Maroon">c<sub>66</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>65</sub></font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E9:112_2"/><i><font color="Green">E112</font></i>: 
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>65</sub></font>,<font color="Maroon">c<sub>66</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="toprealb.html#E3:112_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="toprealb.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E10:112_2"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>65</sub></font>,<font color="Maroon">c<sub>66</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="toprealb.html#E3:112_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="toprealb.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E11:112_2"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>65</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>44</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E8:112_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="toprealb.html#E9:112_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<a NAME="E12:112_2"/><b>then </b>
<a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>65</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>53</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>65</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E13:112_2"/><b>then </b><i><font color="Green">E114</font></i>: 
<a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>65</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>53</sub></font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E11:112_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="toprealb.html#E154"><i><font color="Green">Lemma95</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K15" target="_self">Circle2IntervalL</a>  <a href="tmap_1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K10" target="_self">c[-10]</a> </span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>58</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>63</sub></font> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K15" target="_self">Circle2IntervalL</a>  <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K10" target="_self">c[-10]</a> </span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>58</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>63</sub></font>

<br/>.= 
1 <a href="real_1.html#K3">+</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>65</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#E3:112_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="toprealb.html#E4:112_2"><i><font color="Green">E109</font></i></a>, <a class="txt" href="toprealb.html#E7:112_2"><i><font color="Green">E110</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
1 <a href="real_1.html#K3">+</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>65</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="sin_cos6.html#D4">SIN_COS6:def 4</a></i>
<br/>.= 
1 <a href="real_1.html#K3">+</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="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>65</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan16.html#K1">AffineMap</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>,1</span>)</span> <a href="seq_1.html#K2">.</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>65</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="jordan16.html#D3">JORDAN16:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>51</sub></font> <a href="funct_1.html#K1">.</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>65</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E13:112_2"><i><font color="Green">E114</font></i></a>, <a class="txt" href="toprealb.html#E155"><i><font color="Green">Lemma96</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>51</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>53</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>65</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E11:112_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>51</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>53</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>65</sub></font>,<font color="Maroon">c<sub>66</sub></font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<font color="Maroon">c<sub>51</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>53</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>38</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>63</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E3:112_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
<br/>.= 
<font color="Maroon">c<sub>51</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>53</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>62</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>63</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>51</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>53</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>62</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>63</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E14:112"><i><font color="Green">E104</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>53</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>62</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>63</sub></font>
<b>by </b><i><a class="txt" href="toprealb.html#E14:112"><i><font color="Green">E104</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E23:112"/>
<a href="toprealb.html#K15" target="_self">Circle2IntervalL</a>  <a href="tmap_1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <a href="toprealb.html#K10" target="_self">c[-10]</a> </span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>56</sub></font></span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="toprealb.html#E14:112"><i><font color="Green">E104</font></i></a>, <a class="txt" href="toprealb.html#E21:112"><i><font color="Green">E107</font></i></a>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
