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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="rsspace2.html#NM1" title="RSSPACE2:NM.1">RealHilbertSpace</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">seq</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">seq</font> is  <a href="bhsp_4.html#V2" title="BHSP_4:attr.2">absolutely_summable</a>  holds <br/><font color="Olive" title="b1">seq</font> is  <a href="bhsp_4.html#V1" title="BHSP_4:attr.1">summable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">seq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</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"> ( <font color="Maroon" title="c2">seq</font> is  <a href="bhsp_4.html#V2" title="BHSP_4:attr.2">absolutely_summable</a>  implies <font color="Maroon" title="c2">seq</font> is  <a href="bhsp_4.html#V1" title="BHSP_4:attr.1">summable</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:41"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c2">seq</font> is  <a href="bhsp_4.html#V2" title="BHSP_4:attr.2">absolutely_summable</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">seq</font> is  <a href="bhsp_4.html#V1" title="BHSP_4:attr.1">summable</a> </span><br/>

<a NAME="E2:41"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="bhsp_4.html#D6" target="_self" title="BHSP_4:def.6">Def8</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">k</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>  st <br/> for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</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>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">k</font> holds <br/><span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b3">m</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> )</span><br/><span class="kw">assume </span><a NAME="E1:41_1"/>
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</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>  st <br/> for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</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>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">k</font> holds <br/><span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b3">m</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font></span><br/><a NAME="E2:41_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T215" title="XREAL_1:th.215">XREAL_1:215</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">k</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> <span class="kw"> such that </span><br/><a NAME="E4:41_1"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
 for <font color="Olive" title="b1">m</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>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="series_1.html#T21" title="SERIES_1:th.21">SERIES_1:21</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c5">k</font> = <font color="Maroon" title="c4">k</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">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">k</font> holds <br/><span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">m</font>, <font color="Maroon" title="c7">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"> ( <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">k</font> implies <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> )</span><br/><a NAME="E1:41_1_1"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_4.html#T39" target="_self" title="BHSP_4:th.39">Th41</a></span>;<br/><span class="kw">assume </span><a NAME="E2:41_1_1"/>
( <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font></span><br/><a NAME="E3:41_1_1"/><span class="kw">then </span>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1"><span class="lab"><font color="Green" title="E22">A3</font></span></a></span>;<br/><a NAME="E4:41_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E5:41_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E6:41_1_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p5"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T57" title="COMPLEX1:th.57">COMPLEX1:57</a></span>;<br/><a NAME="E7:41_1_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1_1"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E8:41_1_1"/>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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><span class="kw">hence </span><a NAME="E6:41_1"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">k</font> holds <br/><span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 ; <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>
<span class="kw">hence </span><a NAME="E4:41"/>
<font color="Maroon" title="c2">seq</font> is  <a href="bhsp_4.html#V1" title="BHSP_4:attr.1">summable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_4.html#T10" target="_self" title="BHSP_4:th.10">Th10</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>
