<?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">z</font>, <font color="Maroon" title="c3">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="E1:47"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a> 
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> = <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> $1,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1;<br/>
<a NAME="E2:47"/>
 ex <font color="Olive" title="b1">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font> st <br/>for <font color="Olive" title="b2">k</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>  holds  <font color="Olive" title="b1">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">k</font>)
 
<b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<b>then consider </b><font color="Maroon" title="c4">seq</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:47"/><i><font color="Green" title="E47">A2</font></i>: 
for <font color="Olive" title="b1">k</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>  holds  <font color="Maroon" title="c4">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <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="Olive" title="b1">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font>
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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>thus </b><font color="Maroon" title="c4">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">k</font> = 
<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="c5">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">k</font>
<b>by </b><i><a class="txt" href="lopban_4.html#E4:47"><i><font color="Green" title="E47">A2</font></i></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="c2">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="c5">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="c3">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="c5">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="c2">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">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="c5">k</font></span>)</span>
<b>by </b><i><a class="txt" href="lopban_4.html#E1:47"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="lopban_4.html#T26" target="_self" title="LOPBAN_4:th.26">Th26</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="c2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K7" title="LOPBAN_3:func.7">*</a> <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">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">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"><span class="p4">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">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="c5">k</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D11" title="LOPBAN_3:def.11">LOPBAN_3: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="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K7" title="LOPBAN_3:func.7">*</a> <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">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </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"><span class="p4">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">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/></div><b>end;</b></div>
<a NAME="E6:47"/><b>then </b><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon" title="c4">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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"><font color="Maroon" title="c2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K7" title="LOPBAN_3:func.7">*</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"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<a NAME="E7:47"/><i><font color="Green" title="E49">A4</font></i>: 
 <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="lopban_3.html#D2" title="LOPBAN_3:def.2">LOPBAN_3:def 2</a></i>;<br/>
<a NAME="E8:47"/><i><font color="Green" title="E50">A5</font></i>: 
 <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="lopban_3.html#D2" title="LOPBAN_3:def.2">LOPBAN_3:def 2</a></i>;<br/>
<a NAME="E9:47"/><i><font color="Green" title="E51">A6</font></i>: 
 <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="lopban_3.html#D2" title="LOPBAN_3:def.2">LOPBAN_3:def 2</a></i>;<br/>
<a NAME="E10:47"/><i><font color="Green" title="E52">A7</font></i>: 
( <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"><font color="Maroon" title="c2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K7" title="LOPBAN_3:func.7">*</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"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</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"><font color="Maroon" title="c2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K7" title="LOPBAN_3:func.7">*</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"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</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"><font color="Maroon" title="c2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</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"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="lopban_4.html#E7:47"><i><font color="Green" title="E49">A4</font></i></a>, <a class="txt" href="lopban_4.html#E8:47"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="lopban_4.html#T3" target="_self" title="LOPBAN_4:th.3">Th3</a>, <a class="ref" href="lopban_4.html#T8" target="_self" title="LOPBAN_4:th.8">Th8</a></i>;<br/>
<a NAME="E11:47"/>
( <font color="Maroon" title="c4">seq</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c4">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="txt" href="lopban_4.html#E4:47"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="lopban_4.html#T33" target="_self" title="LOPBAN_4:th.33">Th33</a></i>;<br/>
<b>hence </b><a NAME="E12:47"/>
 <a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="lopban_4.html#K1" title="LOPBAN_4:func.1">ExpSeq</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lopban_4.html#E6:47"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="lopban_4.html#E9:47"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="lopban_4.html#E10:47"><i><font color="Green" title="E52">A7</font></i></a>, <a class="ref" href="lopban_4.html#T1" target="_self" title="LOPBAN_4:th.1">Th1</a></i>;<br/>


</div>
