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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>



<b>assume </b><a NAME="E1:54"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">z</font> is   <a href="real_1.html#NM1">Real</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">z</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1"><b>suppose </b></a><a NAME="E1:54_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then </b><i><font color="Green">E39</font></i>: <font color="Maroon">z</font> = 
the <a href="struct_0.html#U2">Zero</a> of <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>
;<br/><b>thus </b><a NAME="E3:54_1_1"/>
 ex <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="newton.html#K3">|^</a> <font color="Maroon">n</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_1_2_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:54_1_1_2_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<font color="Maroon">n</font> = 
1
<b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="newton.html#K3">|^</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>
;<br/><b>hence </b><a NAME="E3:54_1_1_2_1_1"/>
 ex <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="newton.html#K3">|^</a> <font color="Maroon">n</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:54_1_1_2_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:54_1_1_2_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<font color="Maroon">n</font> = 
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i>, , <a class="ref" href="vectsp_1.html#T95">VECTSP_1:95</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="newton.html#K3">|^</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="uniroots.html#T3" target="_self">Th3</a>, , <a class="ref" href="complfld.html#T9">COMPLFLD:9</a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>
;<br/><b>hence </b><a NAME="E4:54_1_1_2_1_2"/>
 ex <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="newton.html#K3">|^</a> <font color="Maroon">n</font> )
 ;<br/></div><b>end;</b></div></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="54_1_2"><b>suppose </b></a><a NAME="E1:54_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="newton.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font>;<br/><span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,0 = 
 <a href="complex1.html#K6">1r</a> 
<b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="prepower.html#K7">#Z</a> 0
<b>by </b><i><a class="ref" href="prepower.html#T44">PREPOWER:44</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="newton.html#K3">|^</a> 0
<b>by </b><i><a class="ref" href="prepower.html#T46">PREPOWER:46</a></i>
;<br/><a NAME="E3:54_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/><a NAME="E4:54_1_2"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:54_1_2_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<font color="Maroon">n</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K7">#Z</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="prepower.html#T46">PREPOWER:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K7">#Z</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K7">#Z</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T45">PREPOWER:45</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="prepower.html#K7">#Z</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="uniroots.html#T3" target="_self">Th3</a>, <a class="ref" href="prepower.html#T54">PREPOWER:54</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T46">PREPOWER:46</a></i>
;<br/>
<b>hence </b><a NAME="E3:54_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div><a NAME="E5:54_1_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i><a NAME="E6:54_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="newton.html#K3">|^</a> <font color="Maroon">n</font>
 ;<br/><b>hence </b><a NAME="E7:54_1_2"/>
 ex <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="newton.html#K3">|^</a> <font color="Maroon">n</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
