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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>set </b><font color="Maroon">i</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>;<br/>
<b>assume </b><a NAME="E1:25"/><i><font color="Green">E22</font></i>: 
 <a href="sin_cos.html#K20">sin</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 ;<br/>

<b>consider </b><font color="Maroon">w</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:25"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <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> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E4:25"/><i><font color="Green">E24</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E5:25"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> 
 <b>by </b><i><a class="ref" href="complex2.html#T2">COMPLEX2:2</a></i>;<br/>
<a NAME="E6:25"/><i><font color="Green">E32</font></i>: 
0 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>) <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>)
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E19"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E7:25"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">w</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>) <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<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#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>)
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E20"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E8:25"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E17"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="sin_cos.html#T32">SIN_COS:32</a></i>;<br/>
<a NAME="E9:25"/><b>then </b>
<span class="p1">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E10:25"/><b>then </b>
 <a href="sin_cos.html#K23">cos</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E11:25"/><b>then </b>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>) or <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>) )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E18"><i><font color="Green">Lemma53</font></i></a>, <a class="txt" href="sin_cos6.html#E21"><i><font color="Green">Lemma56</font></i></a>, <a class="txt" href="sin_cos6.html#E22"><i><font color="Green">Lemma57</font></i></a>, <a class="txt" href="sin_cos6.html#E15"><i><font color="Green">Lemma50</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:25"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>)
 <b>by </b><i><a class="txt" href="sin_cos6.html#E17"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="complex2.html#T9">COMPLEX2:9</a>, <a class="ref" href="sin_cos.html#T82">SIN_COS:82</a></i>;<br/>


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