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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>assume </b><a NAME="E1:103"/><i><font color="Green">E106</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <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> 1 )
 ;<br/>

<a NAME="E2:103"/><b>then </b>
<span class="p1">(<span class="default"><a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K6" target="_self">arccos</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <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="real_1.html#K3">+</a> 0
 
<b>by </b><i><a class="ref" href="sin_cos6.html#T110" target="_self">Th110</a></i>;<br/>
<a NAME="E3:103"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><a href="sin_cos6.html#K6" target="_self">arccos</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 0 <a href="hidden.html#R1">=</a> <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="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
;<br/>
<a NAME="E4:103"/>
(  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> 1 )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E1:103"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<b>then </b> <a href="sin_cos6.html#K6" target="_self">arccos</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="sin_cos.html#K33">PI</a>  <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K6" target="_self">arccos</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos6.html#T103" target="_self">Th103</a></i>
<br/>.= 
<a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos6.html#E3:103"><i><font color="Green">E107</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>

;<br/>
<b>hence </b><a NAME="E6:103"/>
<span class="p1">(<span class="default"><a href="sin_cos6.html#K6" target="_self">arccos</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="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2
 ;<br/>


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