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<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<i><font color="Green">E67</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> = 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T44" target="_self">Th44</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K13">/</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="sin_cos.html#T37" target="_self">Th37</a></i>
;<br/>
<a NAME="E2:82"/>
<font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5" target="_self">!</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T23">NEWTON:23</a></i>;<br/>
<a NAME="E3:82"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default">0 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E1:82"><i><font color="Green">E67</font></i></a>, <a class="txt" href="sin_cos.html#E67"><i><font color="Green">Lemma50</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:82"/>
<span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span>
 ;<br/>


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