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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="clopban2.html#NM3" title="CLOPBAN2:NM.3">Complex_Banach_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">n</font> being    <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>  holds   <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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K5" title="CLOPBAN4:func.5">Conj</a> (<font color="Olive" title="b3">k</font>,<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K5" title="CLOPBAN4:func.5">Conj</a> (<font color="Olive" title="b3">k</font>,<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">n</font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">n</font> being    <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>  holds   <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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K5" title="CLOPBAN4:func.5">Conj</a> (<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K5" title="CLOPBAN4:func.5">Conj</a> (<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">w</font>)</span>)</span></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">n</font></span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K5" title="CLOPBAN4:func.5">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_0.html#K4" title="NORMSP_0:func.4">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K5" title="CLOPBAN4:func.5">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_0.html#K4" title="NORMSP_0:func.4">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span><br/>









<div><span class="lab"><font color="Green" title="E37">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K5" title="CLOPBAN4:func.5">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_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span><br/><a NAME="E1:38_1"/>
<span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K5" title="CLOPBAN4:func.5">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_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K5" title="CLOPBAN4:func.5">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#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D4" title="NORMSP_0:def.4">NORMSP_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E2:38_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K5" title="CLOPBAN4:func.5">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_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#T105" title="CLVECT_1:th.105">CLVECT_1:105</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:38"/><span class="kw">then </span>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K5" title="CLOPBAN4:func.5">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_0.html#K4" title="NORMSP_0:func.4">.||</a></span> is  <a href="valued_0.html#V9" title="VALUED_0:attr.9">non-decreasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T16" title="SERIES_1:th.16">SERIES_1:16</a></span>;<br/>
<a NAME="E3:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A3</font></span>: 
<span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K5" title="CLOPBAN4:func.5">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_0.html#K4" title="NORMSP_0:func.4">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K5" title="CLOPBAN4:func.5">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_0.html#K4" title="NORMSP_0:func.4">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seqm_3.html#T6" title="SEQM_3:th.6">SEQM_3:6</a></span>;<br/>
<a NAME="E4:38"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
<span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K5" title="CLOPBAN4:func.5">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_0.html#K4" title="NORMSP_0:func.4">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K5" title="CLOPBAN4:func.5">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_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></span>;<br/>
<a NAME="E5:38"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clopban4.html#K5" title="CLOPBAN4:func.5">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_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E37">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:38"/>
 <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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clopban4.html#K5" title="CLOPBAN4:func.5">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_0.html#K4" title="NORMSP_0:func.4">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clopban4.html#K5" title="CLOPBAN4:func.5">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_0.html#K4" title="NORMSP_0:func.4">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
