<?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">k</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="E1:36"/><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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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#K5" target="_self">Alfa</a> <font color="Maroon">k</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">k</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"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K5">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="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#K5" target="_self">Alfa</a> <font color="Maroon">k</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">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"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="clopban3.html#K5">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="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#K5" target="_self">Alfa</a> <font color="Maroon">k</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">k</font></span>)</span>
<b>by </b><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"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="clopban3.html#K5">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="clvect_1.html#K7">-</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#K5" target="_self">Alfa</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></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#D12">CLVECT_1:def 12</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"><span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="clopban3.html#K5">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="clvect_1.html#K7">-</a> <span class="p3">(<span class="default"><a href="clopban4.html#K5" target="_self">Alfa</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="clopban3.html#T20">CLOPBAN3:20</a></i>
;<br/>
<a NAME="E3:36"/>
for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="clopban3.html#K5">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="clvect_1.html#K7">-</a> <span class="p2">(<span class="default"><a href="clopban4.html#K5" target="_self">Alfa</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:36_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<b>thus </b><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="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="clopban3.html#K5">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="clvect_1.html#K7">-</a> <span class="p2">(<span class="default"><a href="clopban4.html#K5" target="_self">Alfa</a> <font color="Maroon">k</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">l</font> = 
<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="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="clopban3.html#K5">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K5" target="_self">Alfa</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D12">CLVECT_1:def 12</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="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K5" target="_self">Alfa</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span>
<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"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <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="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</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#K6" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">l</font>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:36"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="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#K6" target="_self">Conj</a> <font color="Maroon">k</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">k</font>
 <b>by </b><i>, </i>;<br/>


</div>
