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<b>let </b><font color="Maroon" title="c1">s</font>, <font color="Maroon" title="c2">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>set </b><font color="Maroon" title="c3">x</font> =  <a href="sin_cos6.html#K3" title="SIN_COS6:func.3">arcsin</a> <font color="Maroon" title="c2">r</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:81"/><i><font color="Green" title="E92">A1</font></i>: 
<font color="Maroon" title="c1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 <b>and </b><br/><a NAME="E2:81"/><i><font color="Green" title="E93">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<a NAME="E3:81"/><i><font color="Green" title="E94">A3</font></i>: 
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font> &amp; <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E2:81"><i><font color="Green" title="E93">A2</font></i></a>, <a class="txt" href="sin_cos6.html#E13"><i><font color="Green" title="E13">Lm7</font></i></a></i>;<br/>
<a NAME="E4:81"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K3" title="SIN_COS6:func.3">arcsin</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K3" title="SIN_COS6:func.3">arcsin</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#T31" title="SIN_COS:th.31">SIN_COS:31</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><a href="sin_cos6.html#K3" title="SIN_COS6:func.3">arcsin</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  = 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K3" title="SIN_COS6:func.3">arcsin</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>

<br/>.= 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K3" title="SIN_COS6:func.3">arcsin</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D21" title="SIN_COS:def.21">SIN_COS:def 21</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="sin_cos6.html#E3:81"><i><font color="Green" title="E94">A3</font></i></a>, <a class="ref" href="sin_cos6.html#T69" target="_self" title="SIN_COS6:th.69">Th69</a></i>
;<br/>
<a NAME="E6:81"/><b>then </b><i><font color="Green" title="E95">A4</font></i>: 
( <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K3" title="SIN_COS6:func.3">arcsin</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s</font> or <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K3" title="SIN_COS6:func.3">arcsin</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">s</font> )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E2:81"><i><font color="Green" title="E93">A2</font></i></a>, <a class="ref" href="square_1.html#T109" title="SQUARE_1:th.109">SQUARE_1:109</a></i>;<br/>
<a NAME="E7:81"/>
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="sin_cos6.html#K3" title="SIN_COS6:func.3">arcsin</a> <font color="Maroon" title="c2">r</font> &amp;  <a href="sin_cos6.html#K3" title="SIN_COS6:func.3">arcsin</a> <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E3:81"><i><font color="Green" title="E94">A3</font></i></a>, <a class="ref" href="sin_cos6.html#T77" target="_self" title="SIN_COS6:th.77">Th77</a></i>;<br/>
<a NAME="E8:81"/><b>then </b><i><font color="Green" title="E96">A5</font></i>: 
 <a href="sin_cos6.html#K3" title="SIN_COS6:func.3">arcsin</a> <font color="Maroon" title="c2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E9:81"/>
( 0 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">s</font> or <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E1:81"><i><font color="Green" title="E92">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:81"/>
 <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K3" title="SIN_COS6:func.3">arcsin</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">s</font>
 <b>by </b><i><a class="txt" href="sin_cos6.html#E6:81"><i><font color="Green" title="E95">A4</font></i></a>, <a class="txt" href="sin_cos6.html#E8:81"><i><font color="Green" title="E96">A5</font></i></a>, <a class="ref" href="comptrig.html#T28" title="COMPTRIG:th.28">COMPTRIG:28</a>, <a class="ref" href="sin_cos.html#D23" title="SIN_COS:def.23">SIN_COS:def 23</a></i>;<br/>


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