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<b>let </b><font color="Maroon">s</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">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">x</font> =  <a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">r</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:80"/><i><font color="Green">E22</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E2:80"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E3:80"/><i><font color="Green">E24</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="sin_cos6.html#T29" target="_self">Th29</a>, </i>;<br/>
<a NAME="E4:80"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#T31">SIN_COS:31</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  = 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>

<br/>.= 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="sin_cos6.html#T30" target="_self">Th30</a>, <a class="ref" href="sin_cos6.html#T23" target="_self">Th23</a></i>
;<br/>
<a NAME="E6:80"/><b>then </b><i><font color="Green">E31</font></i>: 
( <a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> or <a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="sin_cos6.html#T29" target="_self">Th29</a>, <a class="ref" href="square_1.html#T109">SQUARE_1:109</a></i>;<br/>
<a NAME="E7:80"/>
(  <a href="real_1.html#K1">-</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="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">r</font> &amp;  <a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i><a class="ref" href="sin_cos6.html#T30" target="_self">Th30</a>, <a class="ref" href="sin_cos6.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E8:80"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E9:80"/>
(  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0 or <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="sin_cos6.html#T28" target="_self">Th28</a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/>
<b>hence </b><a NAME="E10:80"/>
 <a href="sin_cos.html#K24">cos</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="sin_cos6.html#T31" target="_self">Th31</a>, <a class="ref" href="sin_cos6.html#T32" target="_self">Th32</a>, <a class="ref" href="comptrig.html#T28">COMPTRIG:28</a>, <a class="ref" href="sin_cos.html#D23">SIN_COS:def 23</a></i>;<br/>


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