<?xml version="1.0"?>
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<a NAME="E1:49"/>
<a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="sin_cos6.html#E24"><i><font color="Green" title="E24">Lm18</font></i></a>, <a class="ref" href="relat_1.html#T104" title="RELAT_1:th.104">RELAT_1:104</a></i>;<br/>
<a NAME="E2:49"/><b>then </b><i><font color="Green" title="E59">A1</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T25" title="RELAT_1:th.25">RELAT_1:25</a></i>;<br/>
<a NAME="E3:49"/><i><font color="Green" title="E60">A2</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <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="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></i>;<br/>
<b>thus </b><a NAME="E4:49"/>
<a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:49_1"/><i><font color="Green" title="E61">A3</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:49_1"/><i><font color="Green" title="E62">A4</font></i>: 
<font color="Maroon" title="c2">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> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> 
 <b>and </b><br/><a NAME="E4:49_1"/><i><font color="Green" title="E63">A5</font></i>: 
<font color="Maroon" title="c2">a</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="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>and </b><br/><a NAME="E5:49_1"/><i><font color="Green" title="E64">A6</font></i>: 
<a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c1">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="sin_cos6.html#E3:49_1"><i><font color="Green" title="E62">A4</font></i></a>, <a class="txt" href="sin_cos6.html#E5:49_1"><i><font color="Green" title="E64">A6</font></i></a></i>;<br/>
<a NAME="E7:49_1"/><i><font color="Green" title="E65">A7</font></i>: 
<a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#D23" title="SIN_COS:def.23">SIN_COS:def 23</a></i>;<br/>
<a NAME="E8:49_1"/><i><font color="Green" title="E66">A8</font></i>: 
(  <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="c4">x</font> &amp; <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E2:49"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="sin_cos6.html#E3:49"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="sin_cos6.html#E1:49_1"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="comptrig.html#T50" title="COMPTRIG:th.50">COMPTRIG:50</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">i</font> = <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>;<br/>
<a NAME="E9:49_1"/><i><font color="Green" title="E67">A9</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>) <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></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/>
<div><i><font color="Green" title="E68">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ;<br/><a NAME="E2:49_1_1"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>)
 <b>by </b><i><a class="txt" href="sin_cos6.html#E5:49_1"><i><font color="Green" title="E64">A6</font></i></a>, <a class="txt" href="sin_cos6.html#E7:49_1"><i><font color="Green" title="E65">A7</font></i></a>, <a class="ref" href="sin_cos6.html#T25" target="_self" title="SIN_COS6:th.25">Th25</a></i>;<br/><a NAME="E3:49_1_1"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>) &amp; <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>) <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>  )
 <b>by </b><i><a class="txt" href="sin_cos6.html#E4:49_1"><i><font color="Green" title="E63">A5</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><a NAME="E4:49_1_1"/><b>then </b>
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  &amp; <span class="p1">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <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>  <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  )
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E5:49_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </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#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></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#K35" title="SIN_COS:func.35">PI</a> </span>)</span> &amp; <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></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#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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#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="E6:49_1_1"/><b>then </b>
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> &amp; <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="sin_cos6.html#E9:49_1"><i><font color="Green" title="E67">A9</font></i></a>, <a class="ref" href="xcmplx_1.html#T188" title="XCMPLX_1:th.188">XCMPLX_1:188</a></i>;<br/><a NAME="E7:49_1_1"/><b>then </b>
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> &amp; <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 <b>by </b><i><a class="ref" href="int_1.html#T21" title="INT_1:th.21">INT_1:21</a>, <a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></i>;<br/><b>hence </b><a NAME="E8:49_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:49_1_2"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>)
 <b>by </b><i><a class="txt" href="sin_cos6.html#E5:49_1"><i><font color="Green" title="E64">A6</font></i></a>, <a class="txt" href="sin_cos6.html#E7:49_1"><i><font color="Green" title="E65">A7</font></i></a>, <a class="ref" href="sin_cos6.html#T26" target="_self" title="SIN_COS6:th.26">Th26</a></i>;<br/><a NAME="E3:49_1_2"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>) &amp; <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>) <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>  )
 <b>by </b><i><a class="txt" href="sin_cos6.html#E4:49_1"><i><font color="Green" title="E63">A5</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><a NAME="E4:49_1_2"/><b>then </b>
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></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#K35" title="SIN_COS:func.35">PI</a> </span>)</span> &amp; <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></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#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <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>  <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#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="E5:49_1_2"/><b>then </b>
(  <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"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> &amp; <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></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#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </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#K35" title="SIN_COS:func.35">PI</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="sin_cos6.html#E9:49_1"><i><font color="Green" title="E67">A9</font></i></a></i>;<br/><a NAME="E6:49_1_2"/><b>then </b>
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> &amp; <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <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#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></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 )
 <b>by </b><i><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a>, <a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></i>;<br/><b>hence </b><a NAME="E7:49_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:49_1"/><b>then </b>
(  <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="c4">x</font> &amp; <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E8:49_1"><i><font color="Green" title="E66">A8</font></i></a>, <a class="txt" href="sin_cos6.html#E10:49_1"><i><font color="Green" title="E68">A10</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E13:49_1"/>
<font color="Maroon" title="c1">x</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c1">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E5:49"/><i><font color="Green" title="E61">A11</font></i>: 
<font color="Maroon" title="c1">a</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c2">a</font> = <font color="Maroon" title="c1">a</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E7:49"/>
(  <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="c2">a</font> &amp; <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E5:49"><i><font color="Green" title="E61">A11</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<a NAME="E8:49"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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></span>)</span>
 
<b>by </b><i><a class="ref" href="comptrig.html#T50" title="COMPTRIG:th.50">COMPTRIG:50</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:49"/><i><font color="Green" title="E62">A12</font></i>: 
<font color="Maroon" title="c3">x</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="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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></span>)</span>
 <b>and </b><br/><a NAME="E11:49"/><i><font color="Green" title="E63">A13</font></i>: 
<span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:49"/><i><font color="Green" title="E64">A14</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="sin_cos6.html#E8"><i><font color="Green" title="E8">Lm2</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="sin_cos6.html#E10:49"><i><font color="Green" title="E62">A12</font></i></a></i>;<br/>
<a NAME="E14:49"/><i><font color="Green" title="E65">A15</font></i>: 
<a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E10:49"><i><font color="Green" title="E62">A12</font></i></a>, <a class="txt" href="sin_cos6.html#E11:49"><i><font color="Green" title="E63">A13</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E15:49"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c4">x</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>  )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E10:49"><i><font color="Green" title="E62">A12</font></i></a>, <a class="txt" href="sin_cos6.html#E12:49"><i><font color="Green" title="E64">A14</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E16:49"/><b>then </b>
( (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c4">x</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  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> or  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> )
 
<b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E17:49"/><b>then </b>
<font color="Maroon" title="c4">x</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="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E5:49"><i><font color="Green" title="E61">A11</font></i></a>, <a class="txt" href="sin_cos6.html#E14:49"><i><font color="Green" title="E65">A15</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a>, <a class="ref" href="sin_cos.html#T33" title="SIN_COS:th.33">SIN_COS:33</a>, <a class="ref" href="sin_cos.html#T81" title="SIN_COS:th.81">SIN_COS:81</a></i>;<br/>
<b>hence </b><a NAME="E18:49"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="sin_cos6.html#E14:49"><i><font color="Green" title="E65">A15</font></i></a>, <a class="txt" href="sin_cos6.html#E8"><i><font color="Green" title="E8">Lm2</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>


</div>
