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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="lopban_2.html#NM3" title="LOPBAN_2:NM.3">Banach_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">z</font>, <font color="Maroon" title="c4">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>







<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a> 
 ;<br/>

<i><font color="Green" title="E39">A2</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="txt" href="lopban_4.html#E1:39"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="lopban_4.html#T25" target="_self" title="LOPBAN_4:th.25">Th25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_4.html#T9" target="_self" title="LOPBAN_4:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="ref" href="normsp_1.html#D6" title="NORMSP_1:def.6">NORMSP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T21" title="LOPBAN_3:th.21">LOPBAN_3:21</a></i>
;<br/>
<a NAME="E3:39"/>
for <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">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" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:39_2"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c5">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">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" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">l</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span> <a href="lopban_3.html#K6" title="LOPBAN_3:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">l</font></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D6" title="NORMSP_1:def.6">NORMSP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K7" title="LOPBAN_4:func.7">Alfa</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">l</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D10" title="LOPBAN_3:def.10">LOPBAN_3:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">l</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lopban_4.html#E1:39_2"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="lopban_4.html#D8" target="_self" title="LOPBAN_4:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">l</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43" title="LOPBAN_3:th.43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">l</font>
<b>by </b><i><a class="txt" href="lopban_4.html#E1:39_2"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="lopban_4.html#D9" target="_self" title="LOPBAN_4:def.9">Def9</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:39"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k</font>
 <b>by </b><i><a class="txt" href="lopban_4.html#E2:39"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="lopban_4.html#T11" target="_self" title="LOPBAN_4:th.11">Th11</a></i>;<br/>


</div>
