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<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>assume </b><a NAME="E1:95"/>
(  <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 )
 ;<br/>

<a NAME="E2:95"/><b>then </b><i><font color="Green">E22</font></i>: 
(  <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">r</font> &amp;  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</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="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<i><font color="Green">E23</font></i>: <font color="Maroon">r</font> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><a href="sin_cos6.html#K6" target="_self">arccos</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></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#K21">sin</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="real_1.html#K4">*</a> <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#K6" target="_self">arccos</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos6.html#T45" target="_self">Th45</a>, <a class="ref" href="sin_cos6.html#T34" target="_self">Th34</a>, <a class="ref" href="sin_cos.html#T82">SIN_COS:82</a></i>
<br/>.= 
 <a href="sin_cos.html#K24">cos</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos6.html#K6" target="_self">arccos</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T4">COMPLEX2:4</a></i>
;<br/>
<a NAME="E4:95"/>
0 <a href="real_1.html#K3">+</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> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K35">PI</a> 
 
<b>by </b><i><a class="ref" href="sin_cos6.html#T45" target="_self">Th45</a>, <a class="ref" href="sin_cos6.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E5:95"/><b>then </b><i><font color="Green">E24</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <a href="sin_cos.html#K35">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> <font color="Maroon">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E6:95"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <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">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sin_cos6.html#T45" target="_self">Th45</a>, <a class="ref" href="sin_cos6.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E7:95"/><b>then </b>
<a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K3">+</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> <font color="Maroon">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E8:95"/><b>then </b>
<a href="sin_cos.html#K35">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> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K35">PI</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<b>hence </b><a NAME="E9:95"/>
 <a href="sin_cos6.html#K6" target="_self">arccos</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K35">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> <font color="Maroon">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sin_cos6.html#T46" target="_self">Th46</a>, , <a class="ref" href="sin_cos6.html#T35" target="_self">Th35</a></i>;<br/>


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