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<b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<i><font color="Green">E20</font></i>:  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> = 
 <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>

<br/>.= 
 <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T158">COMPLEX1:158</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_2"/>
(  <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or  <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1"><b>suppose </b></a><a NAME="E1:27_2_1"/>
 <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_2_1"/>
(  <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> or  <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_2"><b>suppose </b></a><a NAME="E1:27_2_2"/>
 <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:27_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23">cos</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E3:27_2_2"/>
(  <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> or  <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23">cos</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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