<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c2">z</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>assume </b><a NAME="E1:54"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c2">z</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c2">z</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="uniroots.html#E1:54"><i><font color="Green" title="E45">A1</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1"><b>suppose </b></a><a NAME="E1:54_1_1"/><i><font color="Green" title="E46">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:54_1_1"/><b>then </b><i><font color="Green" title="E47">A3</font></i>: 
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/><b>thus </b><a NAME="E3:54_1_1"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> &amp; <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="c2">z</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_1_1_1"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:54_1_1_1_1_1"/><i><font color="Green" title="E48">A4</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>then </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#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c1">n</font> = 
1
<b>by </b><i><a class="ref" href="group_1.html#D8" title="GROUP_1:def.8">GROUP_1:def 8</a>, <a class="ref" href="complfld.html#T10" title="COMPLFLD:th.10">COMPLFLD:10</a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="uniroots.html#E1:54_1_1_1_1_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>
;<br/><b>hence </b><a NAME="E3:54_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> &amp; <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="c2">z</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:54_1_1_1_1_2"/><i><font color="Green" title="E48">A5</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:54_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E49">A6</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<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#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c1">n</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
<b>by </b><i><a class="txt" href="uniroots.html#E2:54_1_1"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="uniroots.html#E1:54_1_1_1_1_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="vectsp_1.html#T95" title="VECTSP_1:th.95">VECTSP_1:95</a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="uniroots.html#E1:54_1_1"><i><font color="Green" title="E46">A2</font></i></a>, <a class="txt" href="uniroots.html#E2:54_1_1_1_1_2"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="complfld.html#T9" title="COMPLFLD:th.9">COMPLFLD:9</a>, <a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></i>
;<br/><b>hence </b><a NAME="E4:54_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> &amp; <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="c2">z</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2"><b>suppose </b></a><a NAME="E1:54_1_2"/><i><font color="Green" title="E46">A7</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>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#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">z</font>,$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> $1;<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#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">z</font>,0 = 
 <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> 
<b>by </b><i><a class="ref" href="group_1.html#D8" title="GROUP_1:def.8">GROUP_1:def 8</a>, <a class="ref" href="complfld.html#T10" title="COMPLFLD:th.10">COMPLFLD:10</a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> 0
<b>by </b><i><a class="ref" href="prepower.html#T44" title="PREPOWER:th.44">PREPOWER:44</a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 0
<b>by </b><i><a class="ref" href="prepower.html#T46" title="PREPOWER:th.46">PREPOWER:46</a></i>
;<br/><a NAME="E3:54_1_2"/><b>then </b><i><font color="Green" title="E47">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/><a NAME="E4:54_1_2"/><i><font color="Green" title="E48">A9</font></i>: 
for <font color="Olive" title="b1">n</font> being   <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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">n</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:54_1_2_2"/><i><font color="Green" title="E49">A10</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">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#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<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="c2">z</font>,<font color="Maroon" title="c4">n</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c2">z</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/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="txt" href="uniroots.html#E1:54_1_2_2"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="prepower.html#T46" title="PREPOWER:th.46">PREPOWER:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T45" title="PREPOWER:th.45">PREPOWER:45</a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E1:54_1_2"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="prepower.html#T54" title="PREPOWER:th.54">PREPOWER:54</a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T46" title="PREPOWER:th.46">PREPOWER:46</a></i>
;<br/>
<b>hence </b><a NAME="E3:54_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 ;<br/>


</div><b>end;</b></div><a NAME="E5:54_1_2"/>
for <font color="Olive" title="b1">n</font> being   <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>  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#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="uniroots.html#E3:54_1_2"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="uniroots.html#E4:54_1_2"><i><font color="Green" title="E48">A9</font></i></a>);<br/></i><a NAME="E6:54_1_2"/><b>then </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#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font>
 ;<br/><b>hence </b><a NAME="E7:54_1_2"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> &amp; <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="c2">z</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
