<?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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <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>assume </b><a NAME="E1:26"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">z</font>,<font color="Maroon">w</font> <a href="clopban4.html#R1" target="_self">are_commutative</a> 
 ;<br/>

<i><font color="Green">E52</font></i>: <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K9">#N</a> 0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="clopban3.html#D13">CLOPBAN3:def 13</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="clopban3.html#D12">CLOPBAN3:def 12</a></i>
;<br/>
<i><font color="Green">E53</font></i>: 0 <a href="binarith.html#K3">-'</a> 0 = 
0 <a href="real_1.html#K5">-</a> 0
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
0

;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K9">#N</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> 0,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 0 = 
<span class="p1">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> 0,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="clopban3.html#D1">CLOPBAN3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8">Coef</a> 0</span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> 0</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> 0</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p3">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K9">*</a> <a href="complex1.html#K6">1r</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> 0</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> 0</span>)</span>
<b>by </b><i>, <a class="ref" href="sin_cos.html#T1">SIN_COS:1</a>, <a class="ref" href="sin_cos.html#D10">SIN_COS:def 10</a>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D13">CLOPBAN3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D13">CLOPBAN3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D12">CLOPBAN3:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D12">CLOPBAN3:def 12</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
;<br/>
<a NAME="E5:26"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E6:26"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:26_4"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
 ;<br/>

<i><font color="Green">E58</font></i>: <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D13">CLOPBAN3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K8">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D12">CLOPBAN3:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i>, <a class="ref" href="clopban3.html#D13">CLOPBAN3:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="clopban3.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="clopban3.html#D9">CLOPBAN3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
;<br/>
<a NAME="E3:26_4"/>
<span class="p1">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_2_1"><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>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D9">CLOPBAN3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D9">CLOPBAN3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D9">CLOPBAN3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="clvect_1.html#D11">CLVECT_1:def 11</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_4_2"/>
<span class="p1">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E59</font></i>: <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="clopban3.html#T19">CLOPBAN3:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D11">CLVECT_1:def 11</a></i>
;<br/>
<a NAME="E5:26_4"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_4"><b>proof </b></a><div class="add">

<i><font color="Green">E61</font></i>: <span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D1">CLOPBAN3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D9">CLOPBAN3:def 9</a></i>
;<br/>
<a NAME="E2:26_4_4"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E6:26_4"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_5"><b>proof </b></a><div class="add">

<i><font color="Green">E71</font></i>: <span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D1">CLOPBAN3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D9">CLOPBAN3:def 9</a></i>
;<br/>
<a NAME="E2:26_4_5"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E7:26_4"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_6"><b>proof </b></a><div class="add">

<a NAME="E1:26_4_6"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:26_4_6"/>
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E3:26_4_6"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="clopban3.html#D9">CLOPBAN3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i>, <a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E8:26_4"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p1">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="clopban4.html#K3" target="_self">Expan</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_7_1"><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><i><font color="Green">E76</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_7_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_4_7_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:26_4_7_1_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_7_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_4_7_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E2:26_4_7_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:26_4_7_1_1_1"/><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E4:26_4_7_1_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E5:26_4_7_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:26_4_7_1_1_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 )
 <b>by </b><i>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>


</div><b>end;</b></div><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D11">CLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D9">CLOPBAN3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></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> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D9">CLOPBAN3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></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> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></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> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
;<br/><b>hence </b><a NAME="E4:26_4_7_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_7_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_4_7_1_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div><i><font color="Green">E109</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_7_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_4_7_1_2_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><b>thus </b><a NAME="E2:26_4_7_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_7_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_4_7_1_2_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<i><font color="Green">E112</font></i>: <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font> = 
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
0

;<br/>
<i><font color="Green">E113</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
0

;<br/>
<a NAME="E4:26_4_7_1_2_1_1"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="sin_cos.html#T1">SIN_COS:1</a></i>;<br/>
<i><font color="Green">E115</font></i>: <span class="p1">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="sin_cos.html#D10">SIN_COS:def 10</a>, <a class="ref" href="sin_cos.html#T1">SIN_COS:1</a></i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
;<br/>
<i><font color="Green">E116</font></i>: <span class="p1">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="sin_cos.html#D10">SIN_COS:def 10</a>, <a class="ref" href="sin_cos.html#T1">SIN_COS:1</a></i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="clvect_1.html#D11">CLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D9">CLOPBAN3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="clopban3.html#D9">CLOPBAN3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , , </i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_7_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_4_7_1_2_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div><i><font color="Green">E118</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_7_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_4_7_1_2_2_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>thus </b><a NAME="E2:26_4_7_1_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_7_1_2_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_4_7_1_2_2_1_1"/><i><font color="Green">E120</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E2:26_4_7_1_2_2_1_1"/><i><font color="Green">E121</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E3:26_4_7_1_2_2_1_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 0
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E4:26_4_7_1_2_2_1_1"/><i><font color="Green">E123</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 0
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E5:26_4_7_1_2_2_1_1"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="sin_cos.html#T1">SIN_COS:1</a></i>;<br/>
<i><font color="Green">E125</font></i>: <span class="p1">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="sin_cos.html#D10">SIN_COS:def 10</a>, <a class="ref" href="sin_cos.html#T1">SIN_COS:1</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="clopban4.html#D1" target="_self">Def1</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
;<br/>
<i><font color="Green">E126</font></i>: <span class="p1">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="sin_cos.html#D10">SIN_COS:def 10</a>, <a class="ref" href="sin_cos.html#T1">SIN_COS:1</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="clopban4.html#D1" target="_self">Def1</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i>, <a class="ref" href="clvect_1.html#D11">CLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D9">CLOPBAN3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> 0</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 0</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> 0</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 0</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> 0</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 0</span>)</span></span>)</span>
<b>by </b><i>, , <a class="txt" href="clopban4.html#E2"><i><font color="Green">Lemma49</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , </i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_7_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_4_7_1_2_2_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:26_4_7_1_2_2_2"/><b>then </b><i><font color="Green">E128</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E3:26_4_7_1_2_2_2"/><b>then </b><i><font color="Green">E129</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E4:26_4_7_1_2_2_2"/><i><font color="Green">E130</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E5:26_4_7_1_2_2_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E6:26_4_7_1_2_2_2"/><b>then </b>
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:26_4_7_1_2_2_2"/><b>then </b>
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:26_4_7_1_2_2_2"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><b>thus </b><a NAME="E9:26_4_7_1_2_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_7_1_2_2_2_1"><b>proof </b></a><div class="add">

<i><font color="Green">E132</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D11">CLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D9">CLOPBAN3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></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> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D9">CLOPBAN3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></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> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E2:26_4_7_1_2_2_2_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_7_1_2_2_2_1_2"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>
<b>hence </b><a NAME="E2:26_4_7_1_2_2_2_1_2"/>
<font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:26_4_7_1_2_2_2_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_7_1_2_2_2_1_3"><b>proof </b></a><div class="add">

<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>
<b>hence </b><a NAME="E2:26_4_7_1_2_2_2_1_3"/>
<font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:26_4_7_1_2_2_2_1"/>
<font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_7_1_2_2_2_1_4"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">k</font>

;<br/>
<b>hence </b><a NAME="E2:26_4_7_1_2_2_2_1_4"/>
<font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font>
 ;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E135</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban4.html#D1" target="_self">Def1</a>, <a class="txt" href="clopban4.html#E2"><i><font color="Green">Lemma49</font></i></a>, , <a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
;<br/>
<i><font color="Green">E136</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="sin_cos.html#D10">SIN_COS:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="sin_cos.html#D10">SIN_COS:def 10</a></i>
;<br/>
<i><font color="Green">E137</font></i>: <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>
<i><font color="Green">E138</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">k</font>,0</span><a href="arytm_0.html#K5">*]</a></span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">k</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span>
<b>by </b><i>, <a class="txt" href="clopban4.html#E3:26"><i><font color="Green">E53</font></i></a>, <a class="ref" href="sin_cos.html#T3">SIN_COS:3</a></i>
;<br/>
<a NAME="E9:26_4_7_1_2_2_2_1"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="sin_cos.html#K4">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K4">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i>, <a class="ref" href="sin_cos.html#T1">SIN_COS:1</a></i>;<br/>
<i><font color="Green">E140</font></i>: <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K8">+</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">k</font>,0</span><a href="arytm_0.html#K5">*]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">k</font>,0</span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 0</span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>

<br/>.= 
<span class="p1"><a href="arytm_0.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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 0</span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
;<br/>
<a NAME="E11:26_4_7_1_2_2_2_1"/>
( <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> &amp; <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">k</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> &amp; 0 <a href="hidden.html#R1">=</a> 0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>;<br/>
<a NAME="E12:26_4_7_1_2_2_2_1"/><b>then </b><i><font color="Green">E141</font></i>: 
(  <a href="complex1.html#K3">Re</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font> &amp;  <a href="complex1.html#K3">Re</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">k</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="complex1.html#K3">Re</a> <a href="complex1.html#K5">0c</a>  <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>;<br/>
<b>then </b><i><font color="Green">E142</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a>, <a class="txt" href="clopban4.html#E6:26"><i><font color="Green">E55</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="sin_cos.html#T3">SIN_COS:3</a></i>
;<br/>
<a NAME="E14:26_4_7_1_2_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">k</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="clopban4.html#E5:26"><i><font color="Green">E54</font></i></a>, <a class="txt" href="clopban4.html#E3"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">k</font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E2:26"><i><font color="Green">E52</font></i></a>, <a class="txt" href="clopban4.html#E2:26_4"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 0</span>)</span></span><a href="arytm_0.html#K5">*]</a></span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E1:26_4"><i><font color="Green">E56</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">0 <a href="nat_1.html#K1">+</a> 0</span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="sin_cos.html#K4">!c</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T1">SIN_COS:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="sin_cos.html#D10">SIN_COS:def 10</a></i>
;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K8">Coef</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban3.html#K9">#N</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="clopban3.html#K9">#N</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E1:26"><i><font color="Green">E51</font></i></a>, <a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
<b>by </b><i>, </i>
;<br/><br/>


</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:26_4_7_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:26_4_7_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:26_4_7_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_4_7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p1">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="clopban4.html#K3" target="_self">Expan</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K9">#N</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="clopban4.html#K2" target="_self">Sift</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K5">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , , , <a class="ref" href="clvect_1.html#D11">CLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="clopban3.html#T19">CLOPBAN3:19</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E7:26"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E8:26"/>
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="clopban3.html#K9">#N</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
 ;<br/>


</div>
