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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="lopban_2.html#NM3">Banach_Algebra</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><a NAME="E1:39"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="lopban_4.html#R1" target="_self">are_commutative</a> 
 ;<br/>

<i><font color="Green">E39</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></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="lopban_3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></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="lopban_3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K7" target="_self">Alfa</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lopban_4.html#E1:39"><i><font color="Green">E38</font></i></a>, <a class="ref" href="lopban_4.html#T25" target="_self">Th25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K7">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K7" target="_self">Alfa</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D10">LOPBAN_3:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="lopban_3.html#K7">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K7" target="_self">Alfa</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_4.html#T9" target="_self">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="lopban_3.html#K7">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K7" target="_self">Alfa</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="normsp_1.html#D6">NORMSP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_3.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">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="lopban_3.html#K7">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K7" target="_self">Alfa</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="lopban_3.html#T21">LOPBAN_3:21</a></i>
;<br/>
<a NAME="E3:39"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="lopban_3.html#K7">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K7" target="_self">Alfa</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K8" target="_self">Conj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:39_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="lopban_3.html#K7">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K7" target="_self">Alfa</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="lopban_3.html#K7">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K7" target="_self">Alfa</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D6">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">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></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="lopban_4.html#K7" target="_self">Alfa</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D10">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">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lopban_4.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">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>5</sub></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">E40</font></i></a>, <a class="ref" href="lopban_4.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></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="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lopban_4.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">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_4.html#K8" target="_self">Conj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="lopban_4.html#E1:39_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="lopban_4.html#D9" target="_self">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="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></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="lopban_3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="lopban_4.html#K8" target="_self">Conj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="lopban_4.html#E2:39"><i><font color="Green">E39</font></i></a>, <a class="ref" href="lopban_4.html#T11" target="_self">Th11</a></i>;<br/>


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