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<b>let </b><font color="Maroon" title="c1">Rseq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a><br/> for <font color="Olive" title="b2">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Olive" title="b1">X</font>  st ( for <font color="Olive" title="b3">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  <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Olive" title="b2">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">Rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font> ) &amp; <font color="Maroon" title="c1">Rseq</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> holds <br/><font color="Olive" title="b2">seq</font> is <a href="bhsp_4.html#V2" title="BHSP_4:attr.2">absolutely_summable</a></span><br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c2">X</font>  st ( for <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  <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Olive" title="b1">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">Rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">n</font> ) &amp; <font color="Maroon" title="c1">Rseq</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> holds <br/><font color="Olive" title="b1">seq</font> is <a href="bhsp_4.html#V2" title="BHSP_4:attr.2">absolutely_summable</a></span><br/><b>let </b><font color="Maroon" title="c3">seq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">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  <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">Rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) &amp; <font color="Maroon" title="c1">Rseq</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> implies <font color="Maroon" title="c3">seq</font> is <a href="bhsp_4.html#V2" title="BHSP_4:attr.2">absolutely_summable</a> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:29"/><i><font color="Green" title="E15">A1</font></i>: 
for <font color="Olive" title="b1">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  <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">Rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 <b>and </b><br/><a NAME="E2:29"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c1">Rseq</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">seq</font> is <a href="bhsp_4.html#V2" title="BHSP_4:attr.2">absolutely_summable</a></span><br/>

<a NAME="E3:29"/>
for <font color="Olive" title="b1">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  <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E1:29_1"/>
<span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span>
 
<b>by </b><i><a class="ref" href="bhsp_2.html#D3" title="BHSP_2:def.3">BHSP_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:29_1"/>
<span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="bhsp_1.html#T34" title="BHSP_1:th.34">BHSP_1:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:29"/><b>then </b>
<span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">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>
 
<b>by </b><i><a class="txt" href="bhsp_4.html#E1:29"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="bhsp_4.html#E2:29"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="series_1.html#T24" title="SERIES_1:th.24">SERIES_1:24</a></i>;<br/>
<b>hence </b><a NAME="E5:29"/>
<font color="Maroon" title="c3">seq</font> is <a href="bhsp_4.html#V2" title="BHSP_4:attr.2">absolutely_summable</a>
 <b>by </b><i><a class="ref" href="bhsp_4.html#D8" target="_self" title="BHSP_4:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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