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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>



<b>assume </b><a NAME="E1:16"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> is   <a href="int_1.html#NM1">Integer</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">a<sub>1</sub></font> is   <a href="int_1.html#NM1">Integer</a>;<br/>
<a NAME="E2:16"/>
<span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,0 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>;<br/>
<a NAME="E3:16"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><b>reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">x</font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">i2</font> = <span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">k</font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i/>;<br/><a NAME="E4:16_1"/>
( <font color="Maroon">i1</font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">i2</font> is   <a href="real_1.html#NM1">Real</a> )
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E5:16_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:16_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1]
 <b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:16"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E6:16"/>
<span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">n</font> is   <a href="int_1.html#NM1">Integer</a>
 ;<br/>


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