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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be  <a href="int_1.html#V1">integer</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:17"/><i><font color="Green">E49</font></i>: 
<a href="sin_cos.html#K33">PI</a>  <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="xxreal_0.html#R1">&lt;</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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
 
<b>by </b><i><a class="ref" href="comptrig.html#T21">COMPTRIG:21</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>assume </b><a NAME="E2:17"/>
( <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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) )
 ;<br/>

<a NAME="E3:17"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>)</span>)</span> <a href="xcmplx_0.html#K6">-</a> <a href="sin_cos.html#K33">PI</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <a href="sin_cos.html#K33">PI</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <a href="sin_cos.html#K33">PI</a>  <a href="xxreal_0.html#R1">&lt;</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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>)</span>)</span> <a href="xcmplx_0.html#K6">-</a> <a href="sin_cos.html#K33">PI</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E4:17"/><b>then </b><i><font color="Green">E50</font></i>: 
( <span class="p1">(<span class="default"><a href="sin_cos.html#K33">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>(<font color="Maroon">c<sub>2</sub></font>) <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <a href="sin_cos.html#K33">PI</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <a href="sin_cos.html#K33">PI</a>  <a href="xxreal_0.html#R1">&lt;</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#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>) )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E1:17"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
 <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> = 
 <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>

<br/>.= 
 <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T34">SIN_COS:34</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T84">SIN_COS:84</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T34">SIN_COS:34</a></i>
;<br/>
<a NAME="E6:17"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E4:17"><i><font color="Green">E50</font></i></a>, <a class="ref" href="sin_cos6.html#T14" target="_self">Th14</a></i>;<br/>
<b>hence </b><a NAME="E7:17"/>
 <a href="sin_cos.html#K23">cos</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>


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