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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>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>









<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:41_1"/>
<span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#D16">CLVECT_1:def 16</a></i>;<br/><b>hence </b><a NAME="E2:41_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="clvect_1.html#T106">CLVECT_1:106</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:41"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> is <a href="seqm_3.html#V3">non-decreasing</a>
 
<b>by </b><i><a class="ref" href="series_1.html#T19">SERIES_1:19</a></i>;<br/>
<a NAME="E3:41"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E4:41"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="clopban4.html#E2:41"><i><font color="Green">E41</font></i></a>, <a class="ref" href="seqm_3.html#T12">SEQM_3:12</a></i>;<br/>
<a NAME="E5:41"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> 0
 
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/>
<a NAME="E6:41"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span> <a href="seq_1.html#K2">.</a> 0
 
<b>by </b><i><a class="txt" href="clopban4.html#E1:41"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:41"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="clvect_1.html#K10">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K6" target="_self">Conj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_1.html#K10">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="clopban4.html#E4:41"><i><font color="Green">E42</font></i></a>, <a class="txt" href="clopban4.html#E5:41"><i><font color="Green">E43</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>


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