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

<b>let </b><font color="Maroon" title="c1">a</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c2">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="c3">k</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>





<b>assume </b><a NAME="E1:58"/><i><font color="Green" title="E47">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 ;<br/>

<a NAME="E2:58"/><b>then </b><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#D3" target="_self" title="PREPOWER:def.3">Def3</a></i>;<br/>
<a NAME="E3:58"/><i><font color="Green" title="E49">A3</font></i>: 
for <font color="Olive" title="b1">m</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="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">m</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">m</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><b>that </b><br/><a NAME="E1:58_1"/><i><font color="Green" title="E50">A4</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>and </b><br/><a NAME="E2:58_1"/><i><font color="Green" title="E51">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">m</font></span>)</span>
 ;<br/>

<a NAME="E3:58_1"/><i><font color="Green" title="E52">A6</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="prepower.html#E1:58"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="prepower.html#T13" target="_self" title="PREPOWER:th.13">Th13</a></i>;<br/>
<b>then </b><i><font color="Green" title="E53">A7</font></i>: <font color="Maroon" title="c4">m</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">m</font></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:58"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="prepower.html#E1:58_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="prepower.html#T34" target="_self" title="PREPOWER:th.34">Th34</a></i>
<br/>.= 
<font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:58"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="prepower.html#E1:58_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="prepower.html#E3:58_1"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="prepower.html#T34" target="_self" title="PREPOWER:th.34">Th34</a></i>
<br/>.= 
<font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <font color="Maroon" title="c1">a</font>
<b>by </b><i><a class="txt" href="prepower.html#E1:58"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="prepower.html#E1:58_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="prepower.html#E32"><i><font color="Green" title="E19">Lm3</font></i></a></i>
;<br/>
<a NAME="E5:58_1"/><i><font color="Green" title="E54">A8</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="prepower.html#E2:58"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="prepower.html#E1:58_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="prepower.html#E32"><i><font color="Green" title="E19">Lm3</font></i></a></i>;<br/>
<a NAME="E6:58_1"/><i><font color="Green" title="E55">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="prepower.html#E2:58"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="prepower.html#T13" target="_self" title="PREPOWER:th.13">Th13</a></i>;<br/>
<a NAME="E7:58_1"/><i><font color="Green" title="E56">A10</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="prepower.html#E1:58"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="prepower.html#E3:58_1"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="prepower.html#D3" target="_self" title="PREPOWER:def.3">Def3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:58_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">m</font></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">m</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="prepower.html#E2:58_1"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_2_1"><b>suppose </b></a><a NAME="E1:58_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">m</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:58_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="prepower.html#E1:58_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="prepower.html#E4:58_1"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="prepower.html#E6:58_1"><i><font color="Green" title="E55">A9</font></i></a>, <a class="txt" href="prepower.html#E32"><i><font color="Green" title="E19">Lm3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_2_2"><b>suppose </b></a><a NAME="E1:58_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">m</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:58_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="prepower.html#E1:58_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="prepower.html#E4:58_1"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="prepower.html#E5:58_1"><i><font color="Green" title="E54">A8</font></i></a>, <a class="txt" href="prepower.html#E7:58_1"><i><font color="Green" title="E56">A10</font></i></a>, <a class="ref" href="prepower.html#T37" target="_self" title="PREPOWER:th.37">Th37</a></i>;<br/></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="58_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:58_2"/>
( <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0 or <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_2_1"><b>suppose </b></a><a NAME="E1:58_2_1"/><i><font color="Green" title="E50">A11</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:58_2_1"/><b>then </b>
 <a href="prepower.html#K1" title="PREPOWER:func.1">abs</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:58_2_1"/><b>then </b><i><font color="Green" title="E51">A12</font></i>: 
 <a href="prepower.html#K1" title="PREPOWER:func.1">abs</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c3">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><a href="prepower.html#K1" title="PREPOWER:func.1">abs</a> <font color="Maroon" title="c3">k</font></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:58_2_1"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="prepower.html#D4" target="_self" title="PREPOWER:def.4">Def4</a></i>
<br/>.= 
<font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><a href="prepower.html#K1" title="PREPOWER:func.1">abs</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E3:58"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="prepower.html#E3:58_2_1"><i><font color="Green" title="E51">A12</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c3">k</font></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:58_2_1"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="prepower.html#D4" target="_self" title="PREPOWER:def.4">Def4</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_2_2"><b>suppose </b></a><a NAME="E1:58_2_2"/><i><font color="Green" title="E50">A13</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:58_2_2"/><b>then </b>
 <a href="prepower.html#K1" title="PREPOWER:func.1">abs</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></i>;<br/><a NAME="E3:58_2_2"/><b>then </b><i><font color="Green" title="E51">A14</font></i>: 
 <a href="prepower.html#K1" title="PREPOWER:func.1">abs</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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/><a NAME="E4:58_2_2"/><i><font color="Green" title="E52">A15</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><a href="prepower.html#K1" title="PREPOWER:func.1">abs</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E1:58"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="prepower.html#T13" target="_self" title="PREPOWER:th.13">Th13</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c3">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><a href="prepower.html#K1" title="PREPOWER:func.1">abs</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 
<b>by </b><i><a class="txt" href="prepower.html#E1:58_2_2"><i><font color="Green" title="E50">A13</font></i></a>, <a class="ref" href="prepower.html#D4" target="_self" title="PREPOWER:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><a href="prepower.html#K1" title="PREPOWER:func.1">abs</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 
<b>by </b><i><a class="txt" href="prepower.html#E3:58"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="prepower.html#E3:58_2_2"><i><font color="Green" title="E51">A14</font></i></a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><a href="prepower.html#K1" title="PREPOWER:func.1">abs</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217" title="XCMPLX_1:th.217">XCMPLX_1:217</a></i>
<br/>.= 
<font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><a href="prepower.html#K1" title="PREPOWER:func.1">abs</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:58"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="prepower.html#E4:58_2_2"><i><font color="Green" title="E52">A15</font></i></a>, <a class="ref" href="prepower.html#T32" target="_self" title="PREPOWER:th.32">Th32</a></i>
<br/>.= 
<font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><a href="prepower.html#K1" title="PREPOWER:func.1">abs</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217" title="XCMPLX_1:th.217">XCMPLX_1:217</a></i>
<br/>.= 
<font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c3">k</font></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:58_2_2"><i><font color="Green" title="E50">A13</font></i></a>, <a class="ref" href="prepower.html#D4" target="_self" title="PREPOWER:def.4">Def4</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_2_3"><b>suppose </b></a><a NAME="E1:58_2_3"/><i><font color="Green" title="E50">A16</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>then </b><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c3">k</font></span>)</span> = 
<font color="Maroon" title="c2">n</font> <a href="prepower.html#K5" title="PREPOWER:func.5">-Root</a> 1
<b>by </b><i><a class="ref" href="prepower.html#T44" target="_self" title="PREPOWER:th.44">Th44</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="prepower.html#E1:58"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="prepower.html#T29" target="_self" title="PREPOWER:th.29">Th29</a></i>
;<br/><b>hence </b><a NAME="E3:58_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K6" title="PREPOWER:func.6">#Z</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E1:58_2_3"><i><font color="Green" title="E50">A16</font></i></a>, <a class="ref" href="prepower.html#T44" target="_self" title="PREPOWER:th.44">Th44</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
