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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="lopban_2.html#NM3" title="LOPBAN_2:NM.3">Banach_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">z</font>, <font color="Maroon" title="c3">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c4">k</font>, <font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>









<div><i><font color="Green" title="E43">A1</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" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:44_1"/>
<span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D10" title="NORMSP_1:def.10">NORMSP_1:def 10</a></i>;<br/><b>hence </b><a NAME="E2:44_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="ref" href="normsp_1.html#T8" title="NORMSP_1:th.8">NORMSP_1:8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:44"/><b>then </b><i><font color="Green" title="E44">A2</font></i>: 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a>
 
<b>by </b><i><a class="ref" href="series_1.html#T19" title="SERIES_1:th.19">SERIES_1:19</a></i>;<br/>
<a NAME="E3:44"/><i><font color="Green" title="E45">A3</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
 
<b>by </b><i><a class="txt" href="lopban_4.html#E2:44"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="seqm_3.html#T12" title="SEQM_3:th.12">SEQM_3:12</a></i>;<br/>
<a NAME="E4:44"/><i><font color="Green" title="E46">A4</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
 
<b>by </b><i><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>;<br/>
<a NAME="E5:44"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
 
<b>by </b><i><a class="txt" href="lopban_4.html#E1:44"><i><font color="Green" title="E43">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:44"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p4">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><a href="lopban_4.html#K8" title="LOPBAN_4:func.8">Conj</a> <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font></span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="lopban_4.html#E3:44"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="lopban_4.html#E4:44"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>


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