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

<b>let </b><font color="Maroon">X</font> be   <a href="clopban2.html#NM3">Complex_Banach_Algebra</a>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>







<b>assume </b><a NAME="E1:42"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>reconsider </b><font color="Maroon">pp</font> = <font color="Maroon">p</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:42"/><i><font color="Green">E52</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:42"/><b>then </b><i><font color="Green">E53</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span>
 
;<br/>
<a NAME="E5:42"/><i><font color="Green">E54</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:42"/>
0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 4 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E7:42"/><b>then </b><i><font color="Green">E55</font></i>: 
0 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E8:42"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:42"/><b>then </b><i><font color="Green">E56</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span>
 
;<br/>
<a NAME="E10:42"/><b>then </b>
0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 4 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E11:42"/><b>then </b><i><font color="Green">E58</font></i>: 
0 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>set </b><font color="Maroon">p1</font> =  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E12:42"/><i><font color="Green">E59</font></i>: 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/>
<a NAME="E13:42"/><i><font color="Green">E60</font></i>: 
(  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> &amp;  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a>, <a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/>
<a NAME="E14:42"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a>  is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T49">SIN_COS:49</a></i>;<br/>
<a NAME="E15:42"/><b>then </b>
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="series_1.html#D2">SERIES_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E17:42"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">k</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">l</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="clopban4.html#D1" target="_self">Def1</a>, <a class="ref" href="comseq_3.html#T4">COMSEQ_3:4</a></i>;<br/>
<a NAME="E18:42"/>
 <a href="clopban3.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> is <a href="clvect_1.html#V6">convergent</a>
 
<b>by </b><i><a class="ref" href="clopban3.html#D2">CLOPBAN3:def 2</a></i>;<br/>
<a NAME="E19:42"/><b>then </b>
for <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font>
 
<b>by </b><i><a class="ref" href="clopban3.html#T4">CLOPBAN3:4</a></i>;<br/>
<a NAME="E20:42"/><b>then </b>
 <a href="clopban3.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> is <a href="csspace3.html#V1">CCauchy</a>
 
<b>by </b><i><a class="ref" href="clopban3.html#T5">CLOPBAN3:5</a></i>;<br/>
<b>then consider </b><font color="Maroon">n2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E22:42"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">k</font>, <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> holds <br/><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">l</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="clopban4.html#D1" target="_self">Def1</a>, <a class="ref" href="csspace3.html#T10">CSSPACE3:10</a></i>;<br/>
<b>set </b><font color="Maroon">n3</font> = <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font>;<br/>
<b>take </b><font color="Maroon">n4</font> = <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span>;<br/>

<div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </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>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:42_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:42_1_1"/><i><font color="Green">E73</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>;<br/><a NAME="E3:42_1_1"/><i><font color="Green">E74</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#T106">CLVECT_1:106</a></i>;<br/><a NAME="E4:42_1_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font>
 <b>by </b><i/>;<br/><a NAME="E5:42_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="txt" href="clopban4.html#E7:42"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font> = 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D16">CLVECT_1:def 16</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="txt" href="clopban4.html#E4:42"><i><font color="Green">E53</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E7:42_1_1"/>
<span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="txt" href="clopban4.html#E5:42"><i><font color="Green">E54</font></i></a>, <a class="txt" href="clopban4.html#E9:42"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:42_1"/>
for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Olive">l</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E76</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:42_2_1"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:42_2_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="txt" href="clopban4.html#E3:42"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E3:42_2_1"/><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#T105">CLVECT_1:105</a></i>;<br/><a NAME="E4:42_2_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:42_2_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:42_2_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:42_2_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E8:42_2_1"/>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="clopban4.html#E9:42"><i><font color="Green">E56</font></i></a>, <a class="txt" href="clopban4.html#E3"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:42_2_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="clopban4.html#E7:42"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:42_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font>
 <b>by </b><i/>;<br/><a NAME="E11:42_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="clopban4.html#E11:42"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E12:42_2_1"/>
<span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="clopban4.html#E5:42"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:42_2"/>
for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:42_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">n4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:42_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E3:42_3"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:42_3"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="clopban4.html#E5:42"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:42_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E6:42_3"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:42_3"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="clopban4.html#E7:42"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:42_3_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font>
 ;<br/><a NAME="E2:42_3_1"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="clopban4.html#E7:42"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:42_3_1"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:42_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E5:42_3_1"/><b>then </b><i><font color="Green">E115</font></i>: 
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:42_3_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">n4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="txt" href="clopban4.html#E5:42"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:42_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="txt" href="clopban4.html#E11:42"><i><font color="Green">E58</font></i></a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><a NAME="E8:42_3_1"/><b>then </b>
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="clopban4.html#T1" target="_self">Th1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:42_3_1"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="txt" href="clopban4.html#E13:42"><i><font color="Green">E60</font></i></a>, <a class="txt" href="clopban4.html#E17:42"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:42_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E11:42_3_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E12:42_3_1"/><b>then </b>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n4</font>
 <b>by </b><i><a class="txt" href="clopban4.html#E17:42"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:42_3_1"/><b>then </b>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="clopban4.html#E5:42"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E14:42_3_1"/><b>then </b><i><font color="Green">E118</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="clopban4.html#E1:42"><i><font color="Green">E51</font></i></a>, <a class="ref" href="clopban4.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E15:42_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font>
 <b>by </b><i/>;<br/><a NAME="E16:42_3_1"/><b>then </b><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E17:42_3_1"/>
<span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="txt" href="clopban4.html#E3:42"><i><font color="Green">E52</font></i></a>, <a class="txt" href="clopban4.html#E12:42"><i><font color="Green">E59</font></i></a></i>;<br/><b>hence </b><a NAME="E18:42_3_1"/>
<span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span>
 <b>by </b><i><a class="ref" href="clopban4.html#T3" target="_self">Th3</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><a NAME="E9:42_3"/><b>then </b><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="comseq_3.html#T7">COMSEQ_3:7</a></i>;<br/><a NAME="E10:42_3"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E11:42_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="clopban4.html#D1" target="_self">Def1</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E12:42_3"/><b>then </b><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="clopban4.html#E11:42"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:42_3"/><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="clopban4.html#E2"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E14:42_3"/><i><font color="Green">E123</font></i>: 
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> &amp; 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 4
<b>by </b><i><a class="txt" href="clopban4.html#E17:42"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>
;<br/><a NAME="E16:42_3"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 4
 <b>by </b><i><a class="txt" href="clopban4.html#E12:42"><i><font color="Green">E59</font></i></a>, <a class="txt" href="clopban4.html#E13:42"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:42_3"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 4 &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2 )
 <b>by </b><i>, <a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/><a NAME="E18:42_3"/><b>then </b>
0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E19:42_3"/><b>then </b><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i><a class="ref" href="clopban4.html#T1" target="_self">Th1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E20:42_3"/>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span>
 <b>by </b><i><a class="txt" href="clopban4.html#E7:42"><i><font color="Green">E55</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E21:42_3"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span>
 ;<br/><a NAME="E22:42_3"/>
for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="clopban4.html#E4:42"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E23:42_3"/><b>then </b><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="clopban4.html#E7:42"><i><font color="Green">E55</font></i></a>, , <a class="ref" href="comseq_3.html#T8">COMSEQ_3:8</a></i>;<br/><a NAME="E24:42_3"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="clopban4.html#E9:42"><i><font color="Green">E56</font></i></a>, <a class="txt" href="clopban4.html#E3"><i><font color="Green">Lemma57</font></i></a></i>;<br/><a NAME="E25:42_3"/><b>then </b><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="clopban4.html#E7:42"><i><font color="Green">E55</font></i></a>, </i>;<br/><a NAME="E26:42_3"/><i><font color="Green">E129</font></i>: 
0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E27:42_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E28:42_3"/><b>then </b><i><font color="Green">E130</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="clopban4.html#T3" target="_self">Th3</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E29:42_3"/><i><font color="Green">E131</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">z</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">pp</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="clopban4.html#E2"><i><font color="Green">Lemma49</font></i></a>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E30:42_3"/><i><font color="Green">E132</font></i>: 
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> &amp; 4 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i/>;<br/><span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">w</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 4
<b>by </b><i><a class="txt" href="clopban4.html#E22:42"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2

;<br/><a NAME="E32:42_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E33:42_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p5">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p5">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="clopban4.html#T2" target="_self">Th2</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E34:42_3"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E26:42"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Olive">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>


</div>
