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

<a NAME="E1:43_1_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Olive">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Olive">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a>  holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:43_1_2_1"/><b>then </b><i><font color="Green">E119</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T131">COMPLEX1:131</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">y</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:43_1_2_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 <b>and </b><br/><a NAME="E4:43_1_2_1"/><i><font color="Green">E121</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:43_1_2_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> 
 <b>and </b><br/><a NAME="E6:43_1_2_1"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 <b>and </b><br/><a NAME="E7:43_1_2_1"/><i><font color="Green">E124</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E8:43_1_2_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> 
 ;<br/>

<a NAME="E9:43_1_2_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> )
 
<b>by </b><i>, , <a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>;<br/>
<a NAME="E10:43_1_2_1"/><b>then </b>
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">y</font></span>)</span> &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">y</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="arytm_0.html#T12">ARYTM_0:12</a></i>;<br/>
<a NAME="E11:43_1_2_1"/><b>then </b>
(  <a href="sin_cos.html#K24">cos</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K24">cos</a> <font color="Maroon">y</font> &amp;  <a href="sin_cos.html#K21">sin</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K21">sin</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>
<a NAME="E12:43_1_2_1"/><b>then </b><i><font color="Green">E126</font></i>: 
(  <a href="sin_cos.html#K24">cos</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K24">cos</a> <font color="Maroon">y</font> &amp;  <a href="sin_cos.html#K21">sin</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>;<br/>
<a NAME="E13:43_1_2_1"/><b>then </b>
(  <a href="sin_cos.html#K24">cos</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> &amp; <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a>, <a class="ref" href="sin_cos.html#D23">SIN_COS:def 23</a></i>;<br/>
<a NAME="E14:43_1_2_1"/><b>then </b><i><font color="Green">E127</font></i>: 
( <a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> &amp; <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="sin_cos.html#D23">SIN_COS:def 23</a></i>;<br/>
<a NAME="E15:43_1_2_1"/><i><font color="Green">E128</font></i>: 
 <a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K22">cos</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="sin_cos.html#D22">SIN_COS:def 22</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1_2_1_1_1"/>
( ( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K35">PI</a>  &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K35">PI</a>  ) or ( <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K35">PI</a>  &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K35">PI</a>  ) or ( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K35">PI</a>  &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K35">PI</a>  ) or ( <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K35">PI</a>  &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K35">PI</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_2_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K35">PI</a>  &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K35">PI</a>  )
 ;<br/><div class="add"><a NAME="E2:43_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span> )
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E3:43_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E129</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K22">cos</a> </span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K22">cos</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="comptrig.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>assume </b><a NAME="E4:43_1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E5:43_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E6:43_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="rfunct_2.html#D3">RFUNCT_2:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_2_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K35">PI</a>  &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K35">PI</a>  )
 ;<br/><div class="add"><a NAME="E2:43_1_2_1_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> )
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E3:43_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E130</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K22">cos</a> </span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K22">cos</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="comptrig.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>assume </b><a NAME="E4:43_1_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E5:43_1_2_1_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E6:43_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="rfunct_2.html#D2">RFUNCT_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_1_1_1_3"><b>suppose </b></a><a NAME="E1:43_1_2_1_1_1_3"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K35">PI</a>  &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K35">PI</a>  )
 ;<br/><div class="add"><a NAME="E2:43_1_2_1_1_1_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E3:43_1_2_1_1_1_3"/><b>then </b>
( <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:43_1_2_1_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_1_1_1_4"><b>suppose </b></a><a NAME="E1:43_1_2_1_1_1_4"/>
( <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K35">PI</a>  &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K35">PI</a>  )
 ;<br/><div class="add"><a NAME="E2:43_1_2_1_1_1_4"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><a href="sin_cos.html#K35">PI</a> ,<span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E3:43_1_2_1_1_1_4"/><b>then </b>
( <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:43_1_2_1_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E17:43_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:43_1_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( ( <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a>  implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( not <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/>


</div>
