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

<b>let </b><font color="Maroon">L</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 ;<br/>

<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_1"/>
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> 1 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> 1 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1"><b>suppose </b></a><a NAME="E1:31_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E3:31_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><a NAME="E4:31_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E5:31_1_1_1"/><i><font color="Green">E53</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_1_1_1_1"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:31_1_1_1_1"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E6:31_1_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:31_1_1_1_2"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:31_1_1_1_2"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:31"><i><font color="Green">E31</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">j</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i>, <a class="txt" href="polynom8.html#E2"><i><font color="Green">Lemma33</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="polynom8.html#T1" target="_self">Th1</a>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_2"><b>suppose </b></a><a NAME="E1:31_1_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> 1
 ;<br/><div class="add"><a NAME="E2:31_1_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:31_1_1_2"/><i><font color="Green">E60</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_1_1_2_1"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:31_1_1_2_1"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E4:31_1_1_2"/><i><font color="Green">E61</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:31_1_1_2_2"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">k</font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E3:31_1_1_2_2"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:31_1_1_2_2"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:31"><i><font color="Green">E31</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p5">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p5">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">x</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p5">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p5">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:31"><i><font color="Green">E31</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="txt" href="polynom8.html#E2"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_3"><b>suppose </b></a><a NAME="E1:31_1_1_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> 1
 ;<br/><div class="add"><a NAME="E2:31_1_1_3"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="vectsp_1.html#T74">VECTSP_1:74</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">j</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="txt" href="polynom8.html#E1:31"><i><font color="Green">E31</font></i></a>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:31"/><i><font color="Green">E64</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1 or <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1"><b>suppose </b></a><a NAME="E1:31_2_1_1"/><i><font color="Green">E65</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="polynom8.html#E1:31"><i><font color="Green">E31</font></i></a>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E3:31_2_1_1"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:31_2_1_1"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2"><b>suppose </b></a><a NAME="E1:31_2_1_2"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:31_2_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> 
 <b>by </b><i/>;<br/><a NAME="E3:31_2_1_2"/>
<font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:31_2_1_2"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:31"><i><font color="Green">E31</font></i></a>, <a class="ref" href="vectsp_1.html#T74">VECTSP_1:74</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:31"/><i><font color="Green">E67</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_3_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font> or  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_1"><b>suppose </b></a><a NAME="E1:31_3_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">k</font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E3:31_3_1_1"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:31_3_1_1"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_3_1_2"><b>suppose </b></a><a NAME="E1:31_3_1_2"/><i><font color="Green">E68</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:31_3_1_2"/><i><font color="Green">E69</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:31_3_1_2"/>
<font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:31_3_1_2"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E2"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="vectsp_1.html#T74">VECTSP_1:74</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:31"/><i><font color="Green">E70</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4"><b>proof </b></a><div class="add">

<a NAME="E1:31_4"/><i><font color="Green">E71</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E2:31_4"/>
<font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:31_4"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E2"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="vectsp_1.html#T74">VECTSP_1:74</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:31"/><i><font color="Green">E72</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_5_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1 or <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_5_1_1"><b>suppose </b></a><a NAME="E1:31_5_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E3:31_5_1_1"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:31_5_1_1"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_5_1_2"><b>suppose </b></a><a NAME="E1:31_5_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:31_5_1_2"/><i><font color="Green">E74</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> 
 <b>by </b><i><a class="ref" href="polynom8.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E3:31_5_1_2"/>
<font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:31_5_1_2"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="vectsp_1.html#T74">VECTSP_1:74</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
;<br/>
<a NAME="E8:31"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1"><i><font color="Green">Lemma29</font></i></a>, <a class="txt" href="polynom8.html#E2"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="vectsp_1.html#T33">VECTSP_1:33</a></i>;<br/>
<b>thus </b> <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> = 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> 
<b>by </b><i><a class="ref" href="polynom8.html#T1" target="_self">Th1</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:31"><i><font color="Green">E31</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,1</span>)</span>

;<br/>


</div>
