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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/>

<a NAME="E1:11"/><i><font color="Green">E13</font></i>: 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="sin_cos.html#T59">SIN_COS:59</a></i>;<br/>
<a NAME="E2:11"/><i><font color="Green">E14</font></i>: 
 <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="sin_cos8.html#E1:11"><i><font color="Green">E13</font></i></a>, <a class="ref" href="sin_cos.html#D27">SIN_COS:def 27</a></i>;<br/>
<a NAME="E3:11"/><i><font color="Green">E15</font></i>: 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="sin_cos.html#T59">SIN_COS:59</a></i>;<br/>
<a NAME="E4:11"/><i><font color="Green">E16</font></i>: 
 <a href="sin_cos.html#K28">exp_R</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="sin_cos8.html#E3:11"><i><font color="Green">E15</font></i></a>, <a class="ref" href="sin_cos.html#D27">SIN_COS:def 27</a></i>;<br/>
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> = 
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D27">SIN_COS:def 27</a></i>
<br/>.= 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D27">SIN_COS:def 27</a></i>
<br/>.= 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos2.html#T12">SIN_COS2:12</a></i>
<br/>.= 
2 <a href="nat_1.html#K2">*</a> 1
<b>by </b><i><a class="ref" href="sin_cos.html#T56">SIN_COS:56</a>, <a class="ref" href="sin_cos.html#D27">SIN_COS:def 27</a>, <a class="ref" href="square_1.html#T83">SQUARE_1:83</a></i>
;<br/>
<b>hence </b><a NAME="E6:11"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</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> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>by </b><i><a class="txt" href="sin_cos8.html#E2:11"><i><font color="Green">E14</font></i></a>, <a class="txt" href="sin_cos8.html#E4:11"><i><font color="Green">E16</font></i></a>, <a class="ref" href="sin_cos2.html#T1">SIN_COS2:1</a></i>;<br/>


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