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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>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="newton.html#K2">|^</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="prepower.html#D1">PREPOWER:def 1</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="prepower.html#T4">PREPOWER:4</a></i>
;<br/>

<b>thus </b><a NAME="E2:77"/>
<font color="Maroon">c<sub>4</sub></font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="newton.html#K2">|^</a> 2
 <b>by </b><i><a class="ref" href="newton.html#T14">NEWTON:14</a></i>;<br/>

<b>thus </b><i><font color="Green">E62</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="newton.html#K2">|^</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D1">PREPOWER:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="prepower.html#T4">PREPOWER:4</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="prepower.html#T4">PREPOWER:4</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>

;<br/>

<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="newton.html#K2">|^</a> 2 = 
<font color="Maroon">c<sub>4</sub></font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="sin_cos.html#E3:77"><i><font color="Green">E62</font></i></a>, <a class="ref" href="newton.html#T13">NEWTON:13</a></i>
;<br/>

<b>thus </b><i><font color="Green">E63</font></i>: <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
1 <a href="prepower.html#K3">|^</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T1">POWER:1</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="newton.html#T15">NEWTON:15</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</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="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> 1
<b>by </b><i><a class="txt" href="sin_cos.html#E5:77"><i><font color="Green">E63</font></i></a></i>
;<br/>


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