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<b>let </b><font color="Maroon" title="c1">e</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><span class="p1">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">e</font>,$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> $1;<br/>
<a NAME="E1:76"/>
<span class="p1">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">e</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">e</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T98" title="GROUP_1:th.98">GROUP_1:98</a></i>;<br/>
<a NAME="E2:76"/><b>then </b><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>;<br/>
<div><i><font color="Green" title="E13">A2</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" title="c2">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>reconsider </b><font color="Maroon" title="c3">n'</font> = <font color="Maroon" title="c2">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">p</font> = <span class="p1">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c3">n'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  ;<br/><b>assume </b><a NAME="E3:76_1"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>]
 ;<br/><span class="p1">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">e</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c4">p</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c1">e</font>
<b>by </b><i><a class="ref" href="group_1.html#D8" title="GROUP_1:def.8">GROUP_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c1">e</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="complfld.html#E3:76_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
;<br/><b>hence </b><a NAME="E5:76_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E4:76"/>
for <font color="Olive" title="b1">n</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S10" title="NAT_1:sch.10">NAT_1:sch 10</a>(<a class="txt" href="complfld.html#E2:76"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="complfld.html#E3:76"><i><font color="Green" title="E13">A2</font></i></a>);</i><br/>


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