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<b>let </b><font color="Maroon">e</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1" target="_self">F_Complex</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1" target="_self">F_Complex</a> </span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">e</font>,<font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> <a href="newton.html#K2">|^</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:76"/>
<span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1" target="_self">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">e</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T98">GROUP_1:98</a></i>;<br/>
<a NAME="E2:76"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>reconsider </b><font color="Maroon">n'</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1" target="_self">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">e</font>,<font color="Maroon">n'</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  ;<br/><b>assume </b><a NAME="E3:76_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1" target="_self">F_Complex</a> </span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">e</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> = 
<font color="Maroon">p</font> <a href="group_1.html#K10">*</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">e</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">e</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/><b>hence </b><a NAME="E5:76_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E4:76"/>
for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S11">NAT_1:sch 11</a>(, );</i><br/>


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