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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="clopban2.html#NM3" title="CLOPBAN2:NM.3">Complex_Banach_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> $1;<br/>
<a NAME="E1:37"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T23" title="NEWTON:th.23">NEWTON:23</a></i>;<br/>
<a NAME="E2:37"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/>
<a NAME="E3:37"/><b>then </b><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E4:37"/><i><font color="Green" title="E38">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:37_1"/><i><font color="Green" title="E39">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>]
 ;<br/>

<a NAME="E2:37_1"/><i><font color="Green" title="E40">A5</font></i>: 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/>
<a NAME="E3:37_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
 
<b>by </b><i><a class="ref" href="clvect_1.html#T106" title="CLVECT_1:th.106">CLVECT_1:106</a></i>;<br/>
<b>hence </b><a NAME="E4:37_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="clopban4.html#E1:37_1"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="clopban4.html#E2:37_1"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="xreal_1.html#T129" title="XREAL_1:th.129">XREAL_1:129</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:37"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="clopban4.html#E3:37"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="clopban4.html#E4:37"><i><font color="Green" title="E38">A3</font></i></a>);<br/></i>
<a NAME="E6:37"/><b>then </b><i><font color="Green" title="E39">A6</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c3">n</font>
 
;<br/>
<a NAME="E7:37"/><i><font color="Green" title="E40">A7</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>;<br/>
<a NAME="E8:37"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
 
<b>by </b><i><a class="txt" href="clopban4.html#E1:37"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="xreal_1.html#T124" title="XREAL_1:th.124">XREAL_1:124</a></i>;<br/>
<a NAME="E9:37"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="clopban4.html#E6:37"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="xreal_1.html#T129" title="XREAL_1:th.129">XREAL_1:129</a></i>;<br/>
<b>hence </b><a NAME="E10:37"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="clopban4.html#E7:37"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="sin_cos.html#D9" title="SIN_COS:def.9">SIN_COS:def 9</a></i>;<br/>


</div>
