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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="clopban2.html#NM3">Complex_Banach_Algebra</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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="E1:44"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="clopban4.html#R1" target="_self">are_commutative</a> 
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>  <a href="subset_1.html#M1">M2</a>(the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>) = <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">a<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:44"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></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">c<sub>4</sub></font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:44"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>4</sub></font> <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="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </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>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<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">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></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="clopban4.html#E4:44"><i><font color="Green">E44</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="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></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="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</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="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E1:44"><i><font color="Green">E43</font></i></a>, <a class="ref" href="clopban4.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="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K6">*</a> <span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></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="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D10">CLOPBAN3:def 10</a></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">c<sub>2</sub></font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K6">*</a> <span class="p3">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="clvect_1.html#D12">CLVECT_1:def 12</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E6:44"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</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"><font color="Maroon">c<sub>2</sub></font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K6">*</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="clvect_1.html#K7">-</a> <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"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="clopban4.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="E7:44"/><i><font color="Green">E46</font></i>: 
 <a href="clopban3.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> is <a href="clvect_1.html#V6">convergent</a>
 
<b>by </b><i><a class="ref" href="clopban3.html#D2">CLOPBAN3:def 2</a></i>;<br/>
<a NAME="E8:44"/><i><font color="Green">E47</font></i>: 
 <a href="clopban3.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> is <a href="clvect_1.html#V6">convergent</a>
 
<b>by </b><i><a class="ref" href="clopban3.html#D2">CLOPBAN3:def 2</a></i>;<br/>
<a NAME="E9:44"/><i><font color="Green">E48</font></i>: 
 <a href="clopban3.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> is <a href="clvect_1.html#V6">convergent</a>
 
<b>by </b><i><a class="ref" href="clopban3.html#D2">CLOPBAN3:def 2</a></i>;<br/>
<a NAME="E10:44"/><i><font color="Green">E49</font></i>: 
( <span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K6">*</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clvect_1.html#K11">lim</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"><font color="Maroon">c<sub>2</sub></font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K6">*</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban4.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="clvect_1.html#K11">lim</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clopban4.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="clvect_1.html#K11">lim</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="clopban4.html#E7:44"><i><font color="Green">E46</font></i></a>, <a class="txt" href="clopban4.html#E8:44"><i><font color="Green">E47</font></i></a>, <a class="ref" href="clopban4.html#T3" target="_self">Th3</a>, <a class="ref" href="clopban4.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E11:44"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clvect_1.html#K11">lim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="clopban4.html#E4:44"><i><font color="Green">E44</font></i></a>, <a class="ref" href="clopban4.html#T32" target="_self">Th32</a></i>;<br/>
<b>thus </b> <a href="clopban3.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span> = 
 <a href="clvect_1.html#K11">lim</a> <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"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D3">CLOPBAN3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clopban4.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="clvect_1.html#K11">lim</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E6:44"><i><font color="Green">E45</font></i></a>, <a class="txt" href="clopban4.html#E9:44"><i><font color="Green">E48</font></i></a>, <a class="txt" href="clopban4.html#E10:44"><i><font color="Green">E49</font></i></a>, <a class="txt" href="clopban4.html#E11:44"><i><font color="Green">E50</font></i></a>, <a class="ref" href="clopban4.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban3.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clopban4.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="clvect_1.html#K11">lim</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D3">CLOPBAN3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban3.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clopban4.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="clopban3.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="clopban4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D3">CLOPBAN3:def 3</a></i>
;<br/>


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