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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="lopban_2.html#NM3">Banach_Algebra</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</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="E1:31"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="lopban_4.html#R1" target="_self">are_commutative</a> 
 ;<br/>

<b>thus </b><span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lopban_3.html#K11">#N</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K5" target="_self">Expan</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lopban_4.html#E1:31"><i><font color="Green">E31</font></i></a>, <a class="ref" href="lopban_4.html#T17" target="_self">Th17</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">c<sub>2</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="normsp_1.html#K6">*</a> <span class="p2">(<span class="default"><a href="lopban_3.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K5" target="_self">Expan</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="normsp_1.html#D8">NORMSP_1:def 8</a></i>
<br/>.= 
<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">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="normsp_1.html#K6">*</a> <span class="p3">(<span class="default"><a href="lopban_4.html#K5" target="_self">Expan</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="lopban_3.html#T24">LOPBAN_3:24</a></i>
<br/>.= 
<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#K6" target="_self">Expan_e</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="lopban_4.html#T18" target="_self">Th18</a></i>
;<br/>


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