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<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="E1:39"/><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>;<br/>

<a NAME="E2:39"/><i><font color="Green">E38</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <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="39_1_1"><b>suppose </b></a><a NAME="E1:39_1_1"/>
<font color="Maroon">i</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">i</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> <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> = 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<font color="Maroon">n</font></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"><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> 

;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2"><b>suppose </b></a><a NAME="E1:39_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:39_1_2"/><i><font color="Green">E42</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">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="39_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:39_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> <font color="Maroon">i</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">i</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:39_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> <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><b>thus </b> <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> = 
<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>,<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><a class="txt" href="polynom8.html#E2:39"><i><font color="Green">E38</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"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom8.html#T1" target="_self">Th1</a></i>
<br/>.= 
 <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">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#T73">VECTSP_1:73</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> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K4">"</a> 
<b>by </b><i><a class="txt" href="polynom8.html#E1:39_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="vectsp_1.html#T73">VECTSP_1:73</a></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">i</font></span>)</span> <a href="vectsp_1.html#K4">"</a> 
<b>by </b><i><a class="txt" href="polynom8.html#E2:39"><i><font color="Green">E38</font></i></a>, </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

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