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<b>set </b><font color="Maroon" title="c21">f</font> =  <a href="toprealb.html#K13" title="TOPREALB:func.13">CircleMap</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> 0</span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">r0</font> = 0 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>set </b><font color="Maroon" title="c23">F</font> =  <a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0;<br/>
<b>reconsider </b><font color="Maroon" title="c24">A</font> =  <a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  ;<br/>
<a NAME="E3:96"/><i><font color="Green" title="E62">A1</font></i>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> 
 
<b>by </b><i><a class="ref" href="toprealb.html#T33" target="_self" title="TOPREALB:th.33">Th33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c25">f</font> =  <a href="toprealb.html#K13" title="TOPREALB:func.13">CircleMap</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> 0</span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c26">Y</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">A</font></span>)</span>;<br/>
<a NAME="E5:96"/><i><font color="Green" title="E63">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E6:96"/><i><font color="Green" title="E64">A3</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c25">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E3:96"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E7:96"/><i><font color="Green" title="E65">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K14" title="TOPREALB:func.14">Circle2IntervalR</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:96"/><i><font color="Green" title="E66">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c27">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:96_1"/><i><font color="Green" title="E67">A6</font></i>: 
<font color="Maroon" title="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K14" title="TOPREALB:func.14">Circle2IntervalR</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c25">f</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c28">b</font> = <font color="Maroon" title="c27">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">A</font></span>)</span> ;<br/><b>consider </b><font color="Maroon" title="c29">x</font>, <font color="Maroon" title="c30">y</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:96_1"/><i><font color="Green" title="E68">A7</font></i>: 
<font color="Maroon" title="c25">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c29">x</font>,<font color="Maroon" title="c30">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>and </b><br/><a NAME="E5:96_1"/><i><font color="Green" title="E69">A8</font></i>: 
( <font color="Maroon" title="c30">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 implies <a href="toprealb.html#K14" title="TOPREALB:func.14">Circle2IntervalR</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c29">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> )
 <b>and </b><br/><a NAME="E6:96_1"/><i><font color="Green" title="E70">A9</font></i>: 
( <font color="Maroon" title="c30">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 implies <a href="toprealb.html#K14" title="TOPREALB:func.14">Circle2IntervalR</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c29">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="toprealb.html#D13" target="_self" title="TOPREALB:def.13">Def13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c31">c</font> = <font color="Maroon" title="c28">b</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><font color="Maroon" title="c25">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">b</font> = 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">b</font>
<b>by </b><i><a class="txt" href="toprealb.html#E5:96"><i><font color="Green" title="E63">A2</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
<b>by </b><i><a class="txt" href="toprealb.html#E72"><i><font color="Green" title="E50">Lm22</font></i></a></i>
;<br/><a NAME="E9:96_1"/><b>then </b>
( <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font> &amp; <font color="Maroon" title="c30">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font> )
 <b>by </b><i><a class="txt" href="toprealb.html#E4:96_1"><i><font color="Green" title="E68">A7</font></i></a>, <a class="ref" href="sppol_2.html#T1" title="SPPOL_2:th.1">SPPOL_2:1</a></i>;<br/><a NAME="E10:96_1"/><b>then </b><i><font color="Green" title="E71">A10</font></i>: 
( <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font></span>)</span> &amp; <font color="Maroon" title="c30">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/><a NAME="E11:96_1"/><b>then </b><i><font color="Green" title="E72">A11</font></i>: 
( <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font></span>)</span> &amp; <font color="Maroon" title="c30">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font></span>)</span> )
 <b>by </b><i><a class="ref" href="sin_cos.html#D21" title="SIN_COS:def.21">SIN_COS:def 21</a>, <a class="ref" href="sin_cos.html#D23" title="SIN_COS:def.23">SIN_COS:def 23</a></i>;<br/><a NAME="E12:96_1"/><i><font color="Green" title="E73">A12</font></i>: 
<span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c31">c</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0
 <b>by </b><i><a class="ref" href="jordan16.html#D3" title="JORDAN16:def.3">JORDAN16:def 3</a></i>;<br/><a NAME="E13:96_1"/><b>then </b><i><font color="Green" title="E74">A13</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">c</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 1
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></i>;<br/><a NAME="E14:96_1"/><i><font color="Green" title="E75">A14</font></i>: 
<span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c31">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c31">c</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0
 <b>by </b><i><a class="ref" href="jordan16.html#D3" title="JORDAN16:def.3">JORDAN16:def 3</a></i>;<br/><a NAME="E15:96_1"/><b>then </b><i><font color="Green" title="E76">A15</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c31">c</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c31">c</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 1
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></i>;<br/><a NAME="E16:96_1"/><i><font color="Green" title="E77">A16</font></i>: 
( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">c</font> &amp; <font color="Maroon" title="c31">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="toprealb.html#E5:96"><i><font color="Green" title="E63">A2</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><div><i><font color="Green" title="E78">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:96_1_2_1"/>
( <font color="Maroon" title="c30">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 or <font color="Maroon" title="c30">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_2_1_1"><b>suppose </b></a><a NAME="E1:96_1_2_1_1"/><i><font color="Green" title="E79">A18</font></i>: 
<font color="Maroon" title="c30">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:96_1_2_1_1"/><b>then </b>
not <span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> ,<span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E10:96_1"><i><font color="Green" title="E71">A10</font></i></a>, <a class="ref" href="comptrig.html#T25" title="COMPTRIG:th.25">COMPTRIG:25</a></i>;<br/><a NAME="E3:96_1_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  or <span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  )
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><a NAME="E4:96_1_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="real_1.html#T73" title="REAL_1:th.73">REAL_1:73</a></i>;<br/><a NAME="E5:96_1_2_1_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">c</font> &amp; <font color="Maroon" title="c31">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 )
 <b>by </b><i><a class="txt" href="toprealb.html#E13:96_1"><i><font color="Green" title="E74">A13</font></i></a>, <a class="txt" href="toprealb.html#E16:96_1"><i><font color="Green" title="E77">A16</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a>, <a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></i>;<br/><a NAME="E6:96_1_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c31">c</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0 &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c31">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><b>hence </b><a href="toprealb.html#K14" title="TOPREALB:func.14">Circle2IntervalR</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E5:96_1"><i><font color="Green" title="E69">A8</font></i></a>, <a class="txt" href="toprealb.html#E11:96_1"><i><font color="Green" title="E72">A11</font></i></a>, <a class="txt" href="toprealb.html#E12:96_1"><i><font color="Green" title="E73">A12</font></i></a>, <a class="txt" href="toprealb.html#E1:96_1_2_1_1"><i><font color="Green" title="E79">A18</font></i></a>, <a class="ref" href="sin_cos6.html#T94" title="SIN_COS6:th.94">SIN_COS6:94</a></i>
<br/>.= 
<font color="Maroon" title="c28">b</font>
<b>by </b><i><a class="txt" href="toprealb.html#E12:96_1"><i><font color="Green" title="E73">A12</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_2_1_2"><b>suppose </b></a><a NAME="E1:96_1_2_1_2"/><i><font color="Green" title="E79">A19</font></i>: 
<font color="Maroon" title="c30">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/><div class="add"><i><font color="Green" title="E80">A20</font></i>: <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c31">c</font></span>)</span></span>)</span> = 
 <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c31">c</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E14:96_1"><i><font color="Green" title="E75">A14</font></i></a>, <a class="ref" href="sin_cos.html#D23" title="SIN_COS:def.23">SIN_COS:def 23</a></i>
<br/>.= 
 <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c31">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T10" title="COMPLEX2:th.10">COMPLEX2:10</a></i>
<br/>.= 
 <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c31">c</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T34" title="SIN_COS:th.34">SIN_COS:34</a></i>
;<br/><a NAME="E3:96_1_2_1_2"/>
not <span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E10:96_1"><i><font color="Green" title="E71">A10</font></i></a>, <a class="txt" href="toprealb.html#E1:96_1_2_1_2"><i><font color="Green" title="E79">A19</font></i></a>, <a class="ref" href="comptrig.html#T24" title="COMPTRIG:th.24">COMPTRIG:24</a></i>;<br/><a NAME="E4:96_1_2_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 or <span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  )
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E5:96_1_2_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">c</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="real_1.html#T73" title="REAL_1:th.73">REAL_1:73</a></i>;<br/><a NAME="E6:96_1_2_1_2"/><b>then </b>
( <font color="Maroon" title="c31">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 or <font color="Maroon" title="c31">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 )
 <b>by </b><i><a class="txt" href="toprealb.html#E13:96_1"><i><font color="Green" title="E74">A13</font></i></a>, <a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></i>;<br/><a NAME="E7:96_1_2_1_2"/><b>then </b>
( 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c31">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> &amp; 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c31">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 )
 <b>by </b><i><a class="txt" href="toprealb.html#E16:96_1"><i><font color="Green" title="E77">A16</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E8:96_1_2_1_2"/><b>then </b><i><font color="Green" title="E81">A21</font></i>: 
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c31">c</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0 &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c31">c</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/> <a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c29">x</font> = 
 <a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c31">c</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E11:96_1"><i><font color="Green" title="E72">A11</font></i></a>, <a class="txt" href="toprealb.html#E12:96_1"><i><font color="Green" title="E73">A12</font></i></a>, <a class="txt" href="toprealb.html#E2:96_1_2_1_2"><i><font color="Green" title="E80">A20</font></i></a>, <a class="ref" href="sin_cos.html#D23" title="SIN_COS:def.23">SIN_COS:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span>,0</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c31">c</font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E14:96_1"><i><font color="Green" title="E75">A14</font></i></a>, <a class="txt" href="toprealb.html#E8:96_1_2_1_2"><i><font color="Green" title="E81">A21</font></i></a>, <a class="ref" href="sin_cos6.html#T94" title="SIN_COS6:th.94">SIN_COS6:94</a></i>
;<br/><b>hence </b><a NAME="E10:96_1_2_1_2"/>
<a href="toprealb.html#K14" title="TOPREALB:func.14">Circle2IntervalR</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">b</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E6:96_1"><i><font color="Green" title="E70">A9</font></i></a>, <a class="txt" href="toprealb.html#E15:96_1"><i><font color="Green" title="E76">A15</font></i></a>, <a class="txt" href="toprealb.html#E1:96_1_2_1_2"><i><font color="Green" title="E79">A19</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K14" title="TOPREALB:func.14">Circle2IntervalR</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">a</font> = 
<a href="toprealb.html#K14" title="TOPREALB:func.14">Circle2IntervalR</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">b</font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E1:96_1"><i><font color="Green" title="E67">A6</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">a</font>
<b>by </b><i><a class="txt" href="toprealb.html#E17:96_1"><i><font color="Green" title="E78">A17</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E10:96"/><b>then </b>
<a href="toprealb.html#K14" title="TOPREALB:func.14">Circle2IntervalR</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c25">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E7:96"><i><font color="Green" title="E65">A4</font></i></a>, <a class="txt" href="toprealb.html#E8:96"><i><font color="Green" title="E66">A5</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E11:96"/><b>then </b>
 <a href="toprealb.html#K14" title="TOPREALB:func.14">Circle2IntervalR</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 
<b>by </b><i><a class="txt" href="toprealb.html#E6:96"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="funct_2.html#T36" title="FUNCT_2:th.36">FUNCT_2:36</a></i>;<br/>
<b>hence </b><a NAME="E12:96"/>
<span class="p1">(<span class="default"><a href="toprealb.html#K13" title="TOPREALB:func.13">CircleMap</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> 0</span>)</span></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="toprealb.html#K14" title="TOPREALB:func.14">Circle2IntervalR</a> 
 <b>by </b><i><a class="txt" href="toprealb.html#E3:96"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="toprealb.html#E6:96"><i><font color="Green" title="E64">A3</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/>


</div>
