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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="E1:76"/>
(  <a href="real_1.html#K1">-</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="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> <a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2 )
 ;<br/>

<a NAME="E2:76"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>1</sub></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#K33">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#K33">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="E3:76"/><i><font color="Green">E89</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="hidden.html#R1">=</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#K33">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#K33">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="txt" href="sin_cos6.html#E7"><i><font color="Green">Lemma7</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>thus </b> <a href="sin_cos6.html#K3" target="_self">arcsin</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20">sin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
 <a href="sin_cos6.html#K3" target="_self">arcsin</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>
<br/>.= 
<a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>

<br/>.= 
<a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="relset_1.html#K8">|</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos6.html#E2:76"><i><font color="Green">E88</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun2.html#K1">id</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="sin_cos6.html#E2:76"><i><font color="Green">E88</font></i></a>, <a class="txt" href="sin_cos6.html#E3:76"><i><font color="Green">E89</font></i></a>, <a class="ref" href="sin_cos6.html#T67" target="_self">Th67</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="sin_cos6.html#E2:76"><i><font color="Green">E88</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>


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