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<b>let </b><font color="Maroon">X</font> be   <a href="lopban_2.html#NM3">Banach_Algebra</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="E1:47"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">z</font>,<font color="Maroon">w</font> <a href="lopban_4.html#R1" target="_self">are_commutative</a> 
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> = <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">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/>
<a NAME="E2:47"/>
 ex <font color="Olive">seq</font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font> st <br/>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>)
 
<b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>then consider </b><font color="Maroon">seq</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:47"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">k</font> <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="Olive">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">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">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> = 
<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">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><a class="txt" href="lopban_4.html#E1:47"><i><font color="Green">E47</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="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">z</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">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="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</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">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="lopban_3.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="lopban_4.html#K1" target="_self">ExpSeq</a> </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="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K8">*</a> <span class="p3">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </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="lopban_3.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="lopban_4.html#K1" target="_self">ExpSeq</a> </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="lopban_3.html#D11">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="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K8">*</a> <span class="p3">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">w</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </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"><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="lopban_4.html#K1" target="_self">ExpSeq</a> </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="normsp_1.html#D6">NORMSP_1:def 6</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E6:47"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">seq</font> <a href="funct_2.html#R2">=</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"><font color="Maroon">z</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K8">*</a> <span class="p2">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K4">-</a> <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"><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<a NAME="E7:47"/><i><font color="Green">E50</font></i>: 
 <a href="lopban_3.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> is <a href="normsp_1.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="lopban_3.html#D2">LOPBAN_3:def 2</a></i>;<br/>
<a NAME="E8:47"/><i><font color="Green">E51</font></i>: 
 <a href="lopban_3.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> is <a href="normsp_1.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="lopban_3.html#D2">LOPBAN_3:def 2</a></i>;<br/>
<a NAME="E9:47"/><i><font color="Green">E53</font></i>: 
 <a href="lopban_3.html#K1">Partial_Sums</a> <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="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> is <a href="normsp_1.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="lopban_3.html#D2">LOPBAN_3:def 2</a></i>;<br/>
<a NAME="E10:47"/><i><font color="Green">E54</font></i>: 
( <span class="p1">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K8">*</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</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"><font color="Maroon">z</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K8">*</a> <span class="p2">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8">lim</a> <span class="p2">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="normsp_1.html#K8">lim</a> <span class="p2">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">w</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="lopban_4.html#E6:47"><i><font color="Green">E49</font></i></a>, <a class="txt" href="lopban_4.html#E7:47"><i><font color="Green">E50</font></i></a>, , </i>;<br/>
<a NAME="E11:47"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">seq</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="txt" href="lopban_4.html#E1:47"><i><font color="Green">E47</font></i></a>, <a class="ref" href="lopban_4.html#D1" target="_self">Def1</a></i>;<br/>
<b>thus </b><a NAME="E12:47"/>
 <a href="lopban_3.html#K2">Sum</a> <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="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lopban_4.html#E4:47"><i><font color="Green">E48</font></i></a>, <a class="txt" href="lopban_4.html#E8:47"><i><font color="Green">E51</font></i></a>, <a class="txt" href="lopban_4.html#E3"><i><font color="Green">Lemma52</font></i></a>, <a class="txt" href="lopban_4.html#E9:47"><i><font color="Green">E53</font></i></a>, </i>;<br/>


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