<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">seq1</font>, <font color="Maroon" title="c3">seq2</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:27"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c2">seq1</font> is <a href="bhsp_4.html#V2" title="BHSP_4:attr.2">absolutely_summable</a>
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c3">seq2</font> is <a href="bhsp_4.html#V2" title="BHSP_4:attr.2">absolutely_summable</a>
 ;<br/>

<a NAME="E3:27"/><i><font color="Green" title="E17">A3</font></i>: 
<span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq1</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#E1:27"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="bhsp_4.html#D8" target="_self" title="BHSP_4:def.8">Def8</a></i>;<br/>
<a NAME="E4:27"/>
<span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq2</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#E2:27"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="bhsp_4.html#D8" target="_self" title="BHSP_4:def.8">Def8</a></i>;<br/>
<a NAME="E5:27"/><b>then </b><i><font color="Green" title="E18">A4</font></i>: 
<span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq1</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq2</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#E3:27"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="series_1.html#T10" title="SERIES_1:th.10">SERIES_1:10</a></i>;<br/>
<a NAME="E6:27"/>
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 <br/> ( <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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>  &amp; <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq1</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq2</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_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> ;<br/>

<b>thus </b><a NAME="E1:27_1"/>
<span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:27_1_1"/>
<span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <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="E2:27_1_1"/>
<span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:27_1"/>
<span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq1</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq2</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2"><b>proof </b></a><div class="add">

<span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> = 
<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"><font color="Maroon" title="c2">seq1</font> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <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/>.= 
<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"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></i>
;<br/>
<a NAME="E2:27_1_2"/><b>then </b>
<span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c2">seq1</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> <a href="real_1.html#K7" title="REAL_1:func.7">+</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">seq2</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_1.html#T36" title="BHSP_1:th.36">BHSP_1:36</a></i>;<br/>
<a NAME="E3:27_1_2"/><b>then </b>
<span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq1</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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">seq2</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/>
<a NAME="E4:27_1_2"/><b>then </b>
<span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq1</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq2</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</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="E5:27_1_2"/>
<span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq1</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><font color="Maroon" title="c3">seq2</font></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E7:27"/><b>then </b>
<span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span></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#E5:27"><i><font color="Green" title="E18">A4</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="E8:27"/>
<font color="Maroon" title="c2">seq1</font> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <font color="Maroon" title="c3">seq2</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>;<br/>


</div>
