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<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<a NAME="E1:8"/><i><font color="Green">E23</font></i>: 
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K20">sin</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>;<br/>
<a NAME="E2:8"/><i><font color="Green">E26</font></i>: 
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K20">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>;<br/>
<a NAME="E3:8"/><i><font color="Green">E29</font></i>: 
<a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K20">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1"><b>suppose </b></a><a NAME="E1:8_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">iN</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E3:8_1_1"/>
 <a href="sin_cos.html#K20">sin</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K20">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">iN</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="sin_cos2.html#T10">SIN_COS2:10</a></i>;<br/><b>hence </b><a NAME="E4:8_1_1"/>
 <a href="sin_cos.html#K20">sin</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K20">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2"><b>suppose </b></a><a NAME="E1:8_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:8_1_2"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><b>then reconsider </b><font color="Maroon">iN</font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon">aa</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E5:8_1_2"/>
 <a href="sin_cos.html#K21">sin</a> <font color="Maroon">aa</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K20">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">iN</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">aa</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="sin_cos2.html#T10">SIN_COS2:10</a></i>;<br/><b>hence </b><a NAME="E6:8_1_2"/>
 <a href="sin_cos.html#K20">sin</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K20">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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