<?xml version="1.0"?>
<div class="add">

<a NAME="E1:106"/><i><font color="Green" title="E88">A1</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  holds  <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>

<a NAME="E1:106_1"/><i><font color="Green" title="E89">A2</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>  holds  <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_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/>

<a NAME="E1:106_1_1"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font>
 
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E2:106_1_1"/>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="ref" href="comseq_3.html#T30" title="COMSEQ_3:th.30">COMSEQ_3:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:106_1"/>
<font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a>  is <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T61" target="_self" title="SIN_COS:th.61">Th61</a></i>;<br/>
<a NAME="E3:106_1"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D11" title="COMSEQ_3:def.11">COMSEQ_3:def 11</a></i>;<br/>
<a NAME="E4:106_1"/><b>then </b><i><font color="Green" title="E90">A3</font></i>: 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a></i>;<br/>
<a NAME="E5:106_1"/>
<font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a>  is <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T61" target="_self" title="SIN_COS:th.61">Th61</a></i>;<br/>
<a NAME="E6:106_1"/><b>then </b><i><font color="Green" title="E91">A4</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a>  is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T64" title="COMSEQ_3:th.64">COMSEQ_3:64</a></i>;<br/>
<a NAME="E7:106_1"/><b>then </b><i><font color="Green" title="E92">A5</font></i>: 
 <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D10" title="COMSEQ_3:def.10">COMSEQ_3:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">PS</font> =  <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span> as  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> <b>by </b><i><a class="txt" href="sin_cos.html#E6:106_1"><i><font color="Green" title="E91">A4</font></i></a>, <a class="ref" href="comseq_3.html#D10" title="COMSEQ_3:def.10">COMSEQ_3:def 10</a></i>;<br/>
<a NAME="E9:106_1"/><i><font color="Green" title="E93">A6</font></i>: 
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c4">PS</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
;<br/>
<a NAME="E10:106_1"/><i><font color="Green" title="E94">A7</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E7:106_1"><i><font color="Green" title="E92">A5</font></i></a>, <a class="ref" href="comseq_2.html#T11" title="COMSEQ_2:th.11">COMSEQ_2:11</a></i>;<br/>
<a NAME="E11:106_1"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E1:106_1"><i><font color="Green" title="E89">A2</font></i></a>, <a class="txt" href="sin_cos.html#E4:106_1"><i><font color="Green" title="E90">A3</font></i></a>, <a class="txt" href="sin_cos.html#E9:106_1"><i><font color="Green" title="E93">A6</font></i></a>, <a class="ref" href="seq_2.html#T32" title="SEQ_2:th.32">SEQ_2:32</a></i>;<br/>
<a NAME="E12:106_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E10:106_1"><i><font color="Green" title="E94">A7</font></i></a>, <a class="ref" href="series_1.html#D3" title="SERIES_1:def.3">SERIES_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:106_1"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:106"/><i><font color="Green" title="E89">A8</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <br/> for <font color="Olive" title="b2">n</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  <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></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"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K2" title="PREPOWER:func.2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon" title="c4">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/>



<i><font color="Green" title="E90">A9</font></i>: <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> = 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">!c</a> </span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="sin_cos.html#D28" target="_self" title="SIN_COS:def.28">Def28</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span>,0</span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="sin_cos.html#T37" target="_self" title="SIN_COS:th.37">Th37</a></i>
;<br/>
<a NAME="E2:106_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <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="sin_cos.html#T42" target="_self" title="SIN_COS:th.42">Th42</a></i>;<br/>
<b>then </b><i><font color="Green" title="E91">A10</font></i>: <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> = 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="comseq_3.html#K2" title="COMSEQ_3:func.2">#N</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E1:106_2"><i><font color="Green" title="E90">A9</font></i></a>, <a class="txt" href="sin_cos.html#E111"><i><font color="Green" title="E84">Lm19</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E91"><i><font color="Green" title="E71">Lm14</font></i></a></i>
;<br/>
<i><font color="Green" title="E92">A11</font></i>: <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></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"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K2" title="PREPOWER:func.2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span> = 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
;<br/>
<a NAME="E5:106_2"/><i><font color="Green" title="E93">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#T42" target="_self" title="SIN_COS:th.42">Th42</a>, <a class="ref" href="newton.html#T18" title="NEWTON:th.18">NEWTON:18</a></i>;<br/>
<a NAME="E6:106_2"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/>
<a NAME="E7:106_2"/><b>then </b>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="power.html#T3" title="POWER:th.3">POWER:3</a></i>;<br/>
<a NAME="E8:106_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E5:106_2"><i><font color="Green" title="E93">A12</font></i></a>, <a class="ref" href="real_2.html#T201" title="REAL_2:th.201">REAL_2:201</a></i>;<br/>
<b>hence </b><a NAME="E9:106_2"/>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></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"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K2" title="PREPOWER:func.2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="sin_cos.html#E3:106_2"><i><font color="Green" title="E91">A10</font></i></a>, <a class="txt" href="sin_cos.html#E4:106_2"><i><font color="Green" title="E92">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon" title="c3">p</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E3:106"/><i><font color="Green" title="E90">A13</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<a NAME="E4:106"/><b>then </b><i><font color="Green" title="E91">A14</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c3">p</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">q</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:106"/><i><font color="Green" title="E92">A15</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 ;<br/>
<a NAME="E8:106"/>
<font color="Maroon" title="c4">p</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E9:106"/><b>then </b><i><font color="Green" title="E93">A16</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="sin_cos.html#E3:106"><i><font color="Green" title="E90">A13</font></i></a>, <a class="txt" href="sin_cos.html#E7:106"><i><font color="Green" title="E92">A15</font></i></a>, <a class="ref" href="real_2.html#T142" title="REAL_2:th.142">REAL_2:142</a></i>;<br/>
<a NAME="E10:106"/><i><font color="Green" title="E94">A17</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">q</font> holds <br/><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:106_3"/><i><font color="Green" title="E95">A18</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">q</font>
 ;<br/>

<a NAME="E2:106_3"/><b>then </b><i><font color="Green" title="E96">A19</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="sin_cos.html#E9:106"><i><font color="Green" title="E93">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:106_3"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 
;<br/>
<a NAME="E4:106_3"/><b>then </b><i><font color="Green" title="E97">A20</font></i>: 
( <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K2" title="PREPOWER:func.2">GeoSeq</a>  is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp;  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K2" title="PREPOWER:func.2">GeoSeq</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="series_1.html#T28" title="SERIES_1:th.28">SERIES_1:28</a></i>;<br/>
<a NAME="E5:106_3"/><b>then </b><i><font color="Green" title="E98">A21</font></i>: 
( <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K2" title="PREPOWER:func.2">GeoSeq</a> </span>)</span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp;  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K2" title="PREPOWER:func.2">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K2" title="PREPOWER:func.2">GeoSeq</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="series_1.html#T13" title="SERIES_1:th.13">SERIES_1:13</a></i>;<br/>
<a NAME="E6:106_3"/><i><font color="Green" title="E99">A22</font></i>: 
for <font color="Olive" title="b1">n</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  <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K2" title="PREPOWER:func.2">GeoSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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/>

<a NAME="E1:106_3_1"/>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></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"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K2" title="PREPOWER:func.2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E2:106"><i><font color="Green" title="E89">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:106_3_1"/>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K2" title="PREPOWER:func.2">GeoSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:106_3"/>
for <font color="Olive" title="b1">n</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  0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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/>

<a NAME="E1:106_3_2"/>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E2:106_3_2"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:106_3"/><b>then </b><i><font color="Green" title="E100">A23</font></i>: 
( <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp;  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K2" title="PREPOWER:func.2">GeoSeq</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="sin_cos.html#E5:106_3"><i><font color="Green" title="E98">A21</font></i></a>, <a class="txt" href="sin_cos.html#E6:106_3"><i><font color="Green" title="E99">A22</font></i></a>, <a class="ref" href="series_1.html#T24" title="SERIES_1:th.24">SERIES_1:24</a></i>;<br/>
<a NAME="E9:106_3"/><i><font color="Green" title="E101">A24</font></i>: 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="prepower.html#K2" title="PREPOWER:func.2">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E4:106_3"><i><font color="Green" title="E97">A20</font></i></a>, <a class="txt" href="sin_cos.html#E5:106_3"><i><font color="Green" title="E98">A21</font></i></a>, <a class="ref" href="xcmplx_1.html#T100" title="XCMPLX_1:th.100">XCMPLX_1:100</a></i>;<br/>
<a NAME="E10:106_3"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3_3"><b>proof </b></a><div class="add">

<a NAME="E1:106_3_3"/><i><font color="Green" title="E102">A25</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E4:106"><i><font color="Green" title="E91">A14</font></i></a>, <a class="txt" href="sin_cos.html#E7:106"><i><font color="Green" title="E92">A15</font></i></a>, <a class="txt" href="sin_cos.html#E1:106_3"><i><font color="Green" title="E95">A18</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E2:106_3_3"/><i><font color="Green" title="E103">A26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E4:106"><i><font color="Green" title="E91">A14</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E3:106_3_3"/><i><font color="Green" title="E104">A27</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">p</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E1:106_3_3"><i><font color="Green" title="E102">A25</font></i></a>, <a class="txt" href="sin_cos.html#E2:106_3_3"><i><font color="Green" title="E103">A26</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E4:106_3_3"/><i><font color="Green" title="E105">A28</font></i>: 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="sin_cos.html#E2:106_3"><i><font color="Green" title="E96">A19</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E5:106_3_3"/><b>then </b>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E3:106_3_3"><i><font color="Green" title="E104">A27</font></i></a>, <a class="ref" href="real_1.html#T73" title="REAL_1:th.73">REAL_1:73</a></i>;<br/>
<b>hence </b><a NAME="E6:106_3_3"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="sin_cos.html#E4:106_3_3"><i><font color="Green" title="E105">A28</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:106_3"/><b>then </b>
(  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">p</font> &amp; <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> )
 
<b>by </b><i><a class="txt" href="sin_cos.html#E1:106"><i><font color="Green" title="E88">A1</font></i></a>, <a class="txt" href="sin_cos.html#E8:106_3"><i><font color="Green" title="E100">A23</font></i></a>, <a class="txt" href="sin_cos.html#E9:106_3"><i><font color="Green" title="E101">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E12:106_3"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c5">q</font>
;<br/>

<b>thus </b><a NAME="E11:106"/>
( <font color="Maroon" title="c5">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; ( for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">q</font> holds <br/><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">p</font> ) )
 <b>by </b><i><a class="txt" href="sin_cos.html#E3:106"><i><font color="Green" title="E90">A13</font></i></a>, <a class="txt" href="sin_cos.html#E4:106"><i><font color="Green" title="E91">A14</font></i></a>, <a class="txt" href="sin_cos.html#E7:106"><i><font color="Green" title="E92">A15</font></i></a>, <a class="txt" href="sin_cos.html#E10:106"><i><font color="Green" title="E94">A17</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/>


</div>
