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<span class="kw">let </span><font color="Maroon" title="c18">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <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#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c18">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>  st <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">A</font> &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c18">A</font> holds <br/><font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c19">f</font> be   <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#K2" title="TOPALG_2:func.2">R^1</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c18">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">A</font> &amp; <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c18">A</font> implies <font color="Maroon" title="c19">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:83"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
<span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">A</font>
 <span class="kw">and </span><br/><a NAME="E2:83"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
<font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c18">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> </span><br/>

<a NAME="E3:83"/><span class="lab"><font color="Green" title="E54">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="toprealb.html#E61"><span class="lab"><font color="Green" title="E42">Lm18</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a>, <a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:83"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>,<a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">f</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c20">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> or <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:83"/><span class="lab"><font color="Green" title="E55">A4</font></span>: 
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">f</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">z</font> = <font color="Maroon" title="c20">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T25" title="PRE_TOPC:th.25">PRE_TOPC:25</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">z1</font> = <font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">z2</font> = <font color="Maroon" title="c21">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> ;<br/>
<a NAME="E7:83"/><span class="lab"><font color="Green" title="E56">A5</font></span>: 
<font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="toprealb.html#T13" target="_self" title="TOPREALB:th.13">Th13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c24">x</font> = <span class="p1">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span>;<br/>
<a NAME="E8:83"/><span class="lab"><font color="Green" title="E57">A6</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="toprealb.html#T13" target="_self" title="TOPREALB:th.13">Th13</a></span>;<br/>
<a NAME="E9:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A7</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="toprealb.html#E72"><span class="lab"><font color="Green" title="E50">Lm22</font></span></a></span>;<br/>
<a NAME="E10:83"/>
<span class="p1">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="toprealb.html#E72"><span class="lab"><font color="Green" title="E50">Lm22</font></span></a></span>;<br/>
<a NAME="E11:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A8</font></span>: 
<span class="p1">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E12:83"/><span class="lab"><font color="Green" title="E60">A9</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c21">z</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_1.html#T29" title="JGRAPH_1:th.29">JGRAPH_1:29</a></span>;<br/>
<a NAME="E13:83"/><span class="lab"><font color="Green" title="E61">A10</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c21">z</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="toprealb.html#T12" target="_self" title="TOPREALB:th.12">Th12</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:83_1"/>
( <font color="Maroon" title="c21">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c21">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1"><span class="kw">suppose </span></a><a NAME="E1:83_1_1"/><span class="lab"><font color="Green" title="E62">A11</font></span>: 
<font color="Maroon" title="c21">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">f</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:83_1_1_1"/>
<span class="p1">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:83_1_1_1"/><span class="kw">then </span>
 <a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:83_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="sin_cos6.html#T96" title="SIN_COS6:th.96">SIN_COS6:96</a></span>;<br/><span class="kw">hence </span><a NAME="E4:83_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:83"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="txt" href="#E12:83"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E1:83_1_1"><span class="lab"><font color="Green" title="E62">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:83_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A12</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</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> <span class="p3">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E4:83_1_1"/>
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</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> <span class="p3">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:83"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E5:83_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A13</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</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> <span class="p3">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1_1"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span></span>)</span> = 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p5">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#T31" target="_self" title="TOPREALB:th.31">Th32</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#D11" target="_self" title="TOPREALB:def.11">Def11</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p0">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T31" title="SIN_COS:th.31">SIN_COS:31</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T31" title="SIN_COS:th.31">SIN_COS:31</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> <span class="p4">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p5">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> <span class="p4">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p5">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="sin_cos6.html#T91" title="SIN_COS6:th.91">SIN_COS6:91</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:83"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="txt" href="#E12:83"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E1:83_1_1"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="ref" href="sin_cos6.html#T103" title="SIN_COS6:th.103">SIN_COS6:103</a></span>
<br/>.= 
<font color="Maroon" title="c20">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:83_1_1"/>
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E3:83"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E5:83_1_1"><span class="lab"><font color="Green" title="E64">A13</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_2"><span class="kw">suppose </span></a><a NAME="E1:83_1_2"/><span class="lab"><font color="Green" title="E62">A14</font></span>: 
<font color="Maroon" title="c21">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">f</font></span><br/><div class="add"><a NAME="E2:83_1_2"/><span class="lab"><font color="Green" title="E63">A15</font></span>: 
<span class="p1">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="txt" href="#E11:83"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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> <span class="p3">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span> = 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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> <span class="p3">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p0">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p0">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealb.html#D11" target="_self" title="TOPREALB:def.11">Def11</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p0">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p4">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="sin_cos6.html#T91" title="SIN_COS6:th.91">SIN_COS6:91</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:83"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="txt" href="#E12:83"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="txt" href="#E1:83_1_2"><span class="lab"><font color="Green" title="E62">A14</font></span></a>, <a class="ref" href="sin_cos6.html#T102" title="SIN_COS6:th.102">SIN_COS6:102</a></span>
<br/>.= 
<font color="Maroon" title="c20">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:83_1_2"/>
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E3:83"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E2:83_1_2"><span class="lab"><font color="Green" title="E63">A15</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
