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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:98"/><i><font color="Green">E63</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T60">SIN_COS:60</a></i>;<br/>
<a NAME="E2:98"/><i><font color="Green">E64</font></i>: 
 <a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="taylor_1.html#T12">TAYLOR_1:12</a></i>;<br/>
<b>assume </b><a NAME="E3:98"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><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>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</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>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 ;<br/>

<a NAME="E4:98"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K24">*</a> 2</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>;<br/>
<a NAME="E5:98"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="taylor_1.html#T4">TAYLOR_1:4</a></i>;<br/>
<a NAME="E6:98"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos7.html#E1:98"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xcmplx_1.html#T114">XCMPLX_1:114</a></i>;<br/>
<a NAME="E7:98"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E8:98"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T80">XCMPLX_1:80</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E9:98"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</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>2</sub></font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos7.html#E8:98"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/>
<a NAME="E10:98"/><i><font color="Green">E67</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&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> 1 )
 
<b>by </b><i><a class="txt" href="sin_cos7.html#E9:98"><i><font color="Green">E66</font></i></a>, <a class="txt" href="sin_cos7.html#E1:98"><i><font color="Green">E63</font></i></a>, <a class="txt" href="sin_cos7.html#E51"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="sin_cos7.html#E52"><i><font color="Green">Lemma52</font></i></a></i>;<br/>
<a NAME="E11:98"/><b>then </b><i><font color="Green">E68</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 4 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="sin_cos7.html#E55"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="sin_cos7.html#E56"><i><font color="Green">Lemma56</font></i></a>, <a class="txt" href="sin_cos7.html#E57"><i><font color="Green">Lemma57</font></i></a>, <a class="txt" href="sin_cos7.html#E58"><i><font color="Green">Lemma58</font></i></a></i>;<br/>
<a NAME="E12:98"/><b>then </b><i><font color="Green">E69</font></i>: 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="sin_cos7.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E13:98"/>
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="sin_cos7.html#E42"><i><font color="Green">Lemma42</font></i></a></i>;<br/>
<a NAME="E14:98"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</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>2</sub></font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos7.html#E9:98"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E15:98"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</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>2</sub></font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
;<br/>
<a NAME="E16:98"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E17:98"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E18:98"/><i><font color="Green">E71</font></i>: 
 <a href="polyeq_1.html#K3">Polynom</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 2</span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="polyeq_1.html#D2">POLYEQ_1:def 2</a></i>;<br/>
<i><font color="Green">E72</font></i>:  <a href="quin_1.html#K1">delta</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 2</span>)</span>,<font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 2</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="quin_1.html#D1">QUIN_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 2</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 2</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>

<br/>.= 
4 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>

;<br/>
<a NAME="E20:98"/>
(  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4">(<span class="default"><a href="binop_2.html#K7">-</a> 2</span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4">(<span class="default"><a href="binop_2.html#K7">-</a> 2</span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sin_cos7.html#E18:98"><i><font color="Green">E71</font></i></a>, <a class="txt" href="sin_cos7.html#E10:98"><i><font color="Green">E67</font></i></a>, <a class="txt" href="sin_cos7.html#E17:98"><i><font color="Green">E70</font></i></a>, <a class="txt" href="sin_cos7.html#E11:98"><i><font color="Green">E68</font></i></a>, <a class="txt" href="sin_cos7.html#E19:98"><i><font color="Green">E72</font></i></a>, <a class="ref" href="quin_1.html#T15">QUIN_1:15</a></i>;<br/>
<a NAME="E21:98"/><b>then </b>
(  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sin_cos7.html#E19:98"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E22:98"/><b>then </b>
(  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sin_cos7.html#E11:98"><i><font color="Green">E68</font></i></a>, <a class="ref" href="square_1.html#T85">SQUARE_1:85</a>, <a class="ref" href="square_1.html#T97">SQUARE_1:97</a></i>;<br/>
<a NAME="E23:98"/><b>then </b>
(  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
;<br/>
<a NAME="E24:98"/><b>then </b>
(  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>;<br/>
<a NAME="E25:98"/><b>then </b>
(  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="sin_cos7.html#E11:98"><i><font color="Green">E68</font></i></a>, <a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>;<br/>
<a NAME="E26:98"/><b>then </b>
(  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 
;<br/>
<a NAME="E27:98"/><b>then </b>
(  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="sin_cos7.html#E11:98"><i><font color="Green">E68</font></i></a>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E28:98"/><b>then </b>
(  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 
;<br/>
<a NAME="E29:98"/><b>then </b>
(  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="sin_cos7.html#E10:98"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>;<br/>
<a NAME="E30:98"/><b>then </b>
(  <a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or  <a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="sin_cos7.html#E12:98"><i><font color="Green">E69</font></i></a>, <a class="txt" href="sin_cos7.html#E2:98"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xcmplx_1.html#T57">XCMPLX_1:57</a></i>;<br/>
<a NAME="E31:98"/><b>then </b>
(  <a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="sin_cos7.html#E2"><i><font color="Green">Lemma2</font></i></a>, <a class="txt" href="sin_cos7.html#E11:98"><i><font color="Green">E68</font></i></a>, <a class="ref" href="power.html#T63">POWER:63</a></i>;<br/>
<b>hence </b><a NAME="E32:98"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 ;<br/>


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