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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">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>let </b><font color="Maroon">k</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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>









<div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_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/><a NAME="E1:44_1"/>
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K8" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</a></i>;<br/><b>hence </b><a NAME="E2:44_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:44"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="normsp_1.html#K7">.||</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="E3:44"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K8" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="normsp_1.html#K7">.||</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="normsp_1.html#K7">||.</a><span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K8" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="lopban_4.html#D1" target="_self">Def1</a>, <a class="ref" href="seqm_3.html#T12">SEQM_3:12</a></i>;<br/>
<a NAME="E4:44"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K8" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="normsp_1.html#K7">.||</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="E5:44"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:44"/>
 <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="normsp_1.html#K7">||.</a><span class="default"><span class="p4">(<span class="default"><a href="lopban_4.html#K8" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</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="normsp_1.html#K7">||.</a><span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K8" target="_self">Conj</a> <font color="Maroon">k</font>,<font color="Maroon">z</font>,<font color="Maroon">w</font></span>)</span></span><a href="normsp_1.html#K7">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="lopban_4.html#E2"><i><font color="Green">Lemma45</font></i></a>, , <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>


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