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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="clopban2.html#NM3" title="CLOPBAN2:NM.3">Complex_Banach_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">z1</font>, <font color="Maroon" title="c3">z2</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="E1:46"/><i><font color="Green" title="E42">A1</font></i>: 
<font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font> <a href="lopban_4.html#R1" title="LOPBAN_4:pred.1">are_commutative</a> 
 ;<br/>

<a NAME="E2:46"/><i><font color="Green" title="E43">A2</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">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a>
 
<b>by </b><i><a class="ref" href="clopban3.html#D2" title="CLOPBAN3:def.2">CLOPBAN3:def 2</a></i>;<br/>
<a NAME="E3:46"/><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c2">z1</font> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span> <a href="clopban3.html#K4" title="CLOPBAN3:func.4">*</a> <font color="Maroon" title="c2">z1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c4">n</font> = 
<font color="Maroon" title="c2">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D8" title="CLOPBAN3:def.8">CLOPBAN3:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c2">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="clopban3.html#K8" title="CLOPBAN3:func.8">#N</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban4.html#D2" target="_self" title="CLOPBAN4:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="clopban3.html#K8" title="CLOPBAN3:func.8">#N</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="clopban3.html#K8" title="CLOPBAN3:func.8">#N</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">z1</font></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E1:46"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="clopban4.html#E3"><i><font color="Green" title="E2">Lm2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="clopban3.html#K8" title="CLOPBAN3:func.8">#N</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">z1</font>
<b>by </b><i><a class="ref" href="clopban3.html#T42" title="CLOPBAN3:th.42">CLOPBAN3:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">z1</font>
<b>by </b><i><a class="ref" href="clopban4.html#D2" target="_self" title="CLOPBAN4:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span> <a href="clopban3.html#K4" title="CLOPBAN3:func.4">*</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c4">n</font>
<b>by </b><i><a class="ref" href="clopban3.html#D9" title="CLOPBAN3:def.9">CLOPBAN3:def 9</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:46_1"/>
<font color="Maroon" title="c2">z1</font> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span> <a href="clopban3.html#K4" title="CLOPBAN3:func.4">*</a> <font color="Maroon" title="c2">z1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><font color="Maroon" title="c2">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <font color="Maroon" title="c3">z2</font></span>)</span> = 
<font color="Maroon" title="c2">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban4.html#D8" target="_self" title="CLOPBAN4:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c2">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K8" title="CLVECT_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">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#D3" title="CLOPBAN3:def.3">CLOPBAN3:def 3</a></i>
<br/>.= 
 <a href="clvect_1.html#K8" title="CLVECT_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</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">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E2:46"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="clopban4.html#T6" target="_self" title="CLOPBAN4:th.6">Th6</a></i>
<br/>.= 
 <a href="clvect_1.html#K8" title="CLVECT_1:func.8">lim</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="c2">z1</font> <a href="clopban3.html#K3" title="CLOPBAN3:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban4.html#T9" target="_self" title="CLOPBAN4:th.9">Th9</a></i>
<br/>.= 
 <a href="clvect_1.html#K8" title="CLVECT_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="c3">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="clopban3.html#K4" title="CLOPBAN3:func.4">*</a> <font color="Maroon" title="c2">z1</font></span>)</span>
<b>by </b><i><a class="txt" href="clopban4.html#E3:46"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="clopban4.html#T9" target="_self" title="CLOPBAN4:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clvect_1.html#K8" title="CLVECT_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">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">z1</font>
<b>by </b><i><a class="txt" href="clopban4.html#E2:46"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="clopban4.html#T7" target="_self" title="CLOPBAN4:th.7">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="clopban4.html#K1" title="CLOPBAN4:func.1">ExpSeq</a> </span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">z1</font>
<b>by </b><i><a class="ref" href="clopban3.html#D3" title="CLOPBAN3:def.3">CLOPBAN3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban4.html#K7" title="CLOPBAN4:func.7">exp</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">z1</font>
<b>by </b><i><a class="ref" href="clopban4.html#D8" target="_self" title="CLOPBAN4:def.8">Def8</a></i>
;<br/>


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