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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">i</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="E1:30"/><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/>

<a NAME="E2:30"/><i><font color="Green">E38</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">L</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#D21">VECTSP_1:def 21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_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="30_1_1"><b>suppose </b></a><a NAME="E1:30_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1_1"/>
(  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> 0 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i><a class="txt" href="polynom8.html#E1"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> 0
 ;<br/><div class="add"><b>hence </b> <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> = 
<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"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> 
<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="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a> 
<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>,<font color="Maroon">i</font></span>)</span> <a href="vectsp_1.html#K4">"</a> 

;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b> <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="group_1.html#K2">1.</a> <font color="Maroon">L</font>
<b>by </b><i/>
<br/>.= 
<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="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="vectsp_1.html#K4">"</a> 
<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"><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>, </i>
;<br/><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="30_1_2"><b>suppose </b></a><a NAME="E1:30_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:30_1_2"/><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> <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="30_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_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="E2:30_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><a NAME="E3:30_1_2"/>
 <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"><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> <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#E1"><i><font color="Green">Lemma29</font></i></a>, </i>;<br/><a NAME="E4:30_1_2"/><b>then </b>
( <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#R1">=</a>  <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> &amp;  <a href="prepower.html#K1">abs</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="txt" href="polynom8.html#E1"><i><font color="Green">Lemma29</font></i></a>, , <a class="ref" href="vectsp_1.html#T73">VECTSP_1:73</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:30_1_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="vectsp_1.html#K4">"</a>  <a href="hidden.html#R1">=</a>  <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>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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