<?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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:40"/><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/>

<b>let </b><font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">j</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1"/>
( ( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 ) or ( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <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> 0 &amp; <font color="Maroon">i</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="40_1_1"><b>suppose </b></a><a NAME="E1:40_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <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">m</font> = <font color="Maroon">i</font>, <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/><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">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span> = 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="polynom8.html#E2"><i><font color="Green">Lemma33</font></i></a></i>
<br/>.= 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">j</font>

;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2"><b>suppose </b></a><a NAME="E1:40_1_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:40_1_2"/><b>then </b>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <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> 0 )
 <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">m</font> = <font color="Maroon">i</font>, <font color="Maroon">n</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="E4:40_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E5:40_1_2"/><i><font color="Green">E42</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">i</font> <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="40_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:40_1_2_1"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:40_1_2_1"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">i</font> <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="E6:40_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">i</font></span>)</span> <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/><a NAME="E7:40_1_2"/><i><font color="Green">E53</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">j</font> <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="40_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:40_1_2_2"/><i><font color="Green">E55</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</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">i</font></span>)</span> <a href="binom.html#K2">|^</a> <span class="p2">(<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> 
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:40_1_2"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<a NAME="E2:40_1_2_2"/>
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">i</font></span>)</span> <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> <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="E3:40_1_2_2"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">j</font> <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="polynom8.html#T2" target="_self">Th2</a>, <a class="ref" href="vectsp_1.html#T74">VECTSP_1:74</a></i>;<br/>


</div><b>end;</b></div><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">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></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">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> 
<b>by </b><i><a class="txt" href="polynom8.html#E4:40_1_2"><i><font color="Green">E39</font></i></a>, , </i>
<br/>.= 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="polynom8.html#E2"><i><font color="Green">Lemma33</font></i></a></i>
<br/>.= 
 <a href="polynom8.html#K3" target="_self">pow</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">i</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">n</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</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">i</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="vectsp_1.html#K4">"</a> 
<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> <span class="p3">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K4">"</a> 
<b>by </b><i>, </i>
<br/>.= 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="polynom8.html#E5:40_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="vectsp_1.html#T73">VECTSP_1:73</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_3"><b>suppose </b></a><a NAME="E1:40_1_3"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:40_1_3"/><b>then </b>
(  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="xcmplx_0.html#K4">-</a> 0 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <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">m</font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font>, <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="E4:40_1_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span>
 ;<br/><a NAME="E5:40_1_3"/><i><font color="Green">E60</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/><a NAME="E6:40_1_3"/><i><font color="Green">E61</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">i</font> <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="40_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:40_1_3_1"/><i><font color="Green">E62</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">i</font> <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> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:40_1_3_1"/>
<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">i</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/>
<b>hence </b><a NAME="E3:40_1_3_1"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">i</font> <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><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">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></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">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> 
<b>by </b><i>, , </i>
<br/>.= 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="polynom8.html#E2"><i><font color="Green">Lemma33</font></i></a></i>
<br/>.= 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">n</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="polynom8.html#K3" target="_self">pow</a> <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>,<font color="Maroon">i</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">j</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#T73">VECTSP_1:73</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_4"><b>suppose </b></a><a NAME="E1:40_1_4"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:40_1_4"/><b>then </b>
(  <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> 0 &amp;  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="xcmplx_0.html#K4">-</a> 0 )
 <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">m</font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font>, <font color="Maroon">n</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="E4:40_1_4"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E5:40_1_4"/><i><font color="Green">E65</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">i</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="40_1_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:40_1_4_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">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:40_1_4_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">i</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="E6:40_1_4"/><i><font color="Green">E66</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">i</font> <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="40_1_4_2"><b>proof </b></a><div class="add">

<a NAME="E1:40_1_4_2"/><i><font color="Green">E67</font></i>: 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</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="binom.html#K2">|^</a> <span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:40_1_4_2"/>
<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/>
<a NAME="E3:40_1_4_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="binom.html#K2">|^</a> <span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">i</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/>;<br/>
<b>hence </b><a NAME="E4:40_1_4_2"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">i</font> <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><a NAME="E7:40_1_4"/><i><font color="Green">E68</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> <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span> = 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">n</font>
<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> <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">i</font></span>)</span></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="vectsp_1.html#K4">"</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</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">i</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="vectsp_1.html#K4">"</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="vectsp_1.html#K4">"</a> 
<b>by </b><i>, </i>
<br/>.= 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">j</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <span class="p2">(<span class="default"><span class="p3">(<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>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">j</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#T73">VECTSP_1:73</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
