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<b>let </b><font color="Maroon">X</font> be   <a href="clopban2.html#NM3">Complex_Banach_Algebra</a>;<br/><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>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="E1:28"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">z</font>,<font color="Maroon">w</font> <a href="clopban4.html#R1" target="_self">are_commutative</a> 
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1">*</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="clopban3.html#K9">#N</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K1">*</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"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</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">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="clvect_1.html#D14">CLVECT_1:def 14</a></i>
<br/>.= 
<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"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4">!c</a> </span>)</span></span>)</span> <a href="clvect_1.html#K9">*</a> <span class="p3">(<span class="default"><a href="clopban4.html#K3" target="_self">Expan</a> <font color="Maroon">n</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="clopban3.html#T23">CLOPBAN3:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clopban3.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="clopban4.html#K4" target="_self">Expan_e</a> <font color="Maroon">n</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">n</font>
<b>by </b><i/>
;<br/>


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