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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">z1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">z2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:51"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="lopban_4.html#R1" target="_self">are_commutative</a> 
 ;<br/>

<a NAME="E2:51"/><i><font color="Green">E48</font></i>: 
 <a href="lopban_3.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">z2</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="E3:51"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">z1</font> <a href="lopban_3.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="lopban_3.html#K7">*</a> <font color="Maroon">z1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="lopban_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">z2</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">n</font> = 
<font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D9">LOPBAN_3:def 9</a></i>
<br/>.= 
<font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="lopban_3.html#K11">#N</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="lopban_3.html#K11">#N</a> <font color="Maroon">n</font></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">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="lopban_3.html#K11">#N</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z1</font></span>)</span>
<b>by </b><i><a class="txt" href="lopban_4.html#E3:51"><i><font color="Green">E49</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="lopban_3.html#K11">#N</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z1</font>
<b>by </b><i><a class="ref" href="lopban_3.html#T43">LOPBAN_3:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z1</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="lopban_3.html#K7">*</a> <font color="Maroon">z1</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="lopban_3.html#D10">LOPBAN_3:def 10</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:51_1"/>
<font color="Maroon">z1</font> <a href="lopban_3.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span> <a href="lopban_3.html#K7">*</a> <font color="Maroon">z1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><font color="Maroon">z1</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="lopban_4.html#K10" target="_self">exp</a> <font color="Maroon">z2</font></span>)</span> = 
<font color="Maroon">z1</font> <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">z2</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#E1:51"><i><font color="Green">E47</font></i></a></i>
<br/>.= 
 <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="lopban_3.html#K6">*</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">z2</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="normsp_1.html#K8">lim</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">z1</font> <a href="lopban_3.html#K6">*</a> <span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <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">z2</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="lopban_3.html#K7">*</a> <font color="Maroon">z1</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_3.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="lopban_4.html#K1" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z1</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_4.html#K10" target="_self">exp</a> <font color="Maroon">z2</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z1</font>
<b>by </b><i><a class="txt" href="lopban_4.html#E1:51"><i><font color="Green">E47</font></i></a></i>
;<br/>


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