<?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">seq</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>set </b><font color="Maroon" title="c3">PSseq</font> =  <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font>;<br/>
<b>assume </b><a NAME="E1:11"/>
<font color="Maroon" title="c2">seq</font> is <a href="bhsp_4.html#V1" title="BHSP_4:attr.1">summable</a>
 ;<br/>

<a NAME="E2:11"/><b>then </b><i><font color="Green" title="E7">A1</font></i>: 
 <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a>
 
<b>by </b><i><a class="ref" href="bhsp_4.html#D2" target="_self" title="BHSP_4:def.2">Def2</a></i>;<br/>
<a NAME="E3:11"/><b>then </b><i><font color="Green" title="E8">A2</font></i>: 
( <span class="p1">(<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="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1 is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a> &amp;  <a href="bhsp_2.html#K1" title="BHSP_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<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="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_2.html#K1" title="BHSP_2:func.1">lim</a> <span class="p1">(<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> )
 
<b>by </b><i><a class="ref" href="bhsp_3.html#T44" title="BHSP_3:th.44">BHSP_3:44</a></i>;<br/>
<b>then </b><i><font color="Green" title="E9">A3</font></i>:  <a href="bhsp_2.html#K1" title="BHSP_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1</span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<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></span>)</span> = 
<span class="p1">(<span class="default"><a href="bhsp_2.html#K1" title="BHSP_2:func.1">lim</a> <span class="p2">(<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></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="bhsp_2.html#K1" title="BHSP_2:func.1">lim</a> <span class="p2">(<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></span>)</span>
<b>by </b><i><a class="txt" href="bhsp_4.html#E2:11"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="bhsp_2.html#T14" title="BHSP_2:th.14">BHSP_2:14</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">X</font>)
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
;<br/>
<a NAME="E5:11"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1 <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1</span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1"><b>now </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/><span class="p1">(<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#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<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#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="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_4.html#D1" target="_self" title="BHSP_4:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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#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="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_3.html#D5" title="BHSP_3:def.5">BHSP_3:def 5</a></i>
;<br/><b>hence </b><a NAME="E2:11_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<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="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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#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="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="bhsp_3.html#D5" title="BHSP_3:def.5">BHSP_3:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:11_2"/><b>then </b><i><font color="Green" title="E11">A5</font></i>: 
<span class="p1">(<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="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1 <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<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="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1</span>)</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="E3:11_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1</span>)</span> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<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#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<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></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_2_1"><b>now </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><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1</span>)</span> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<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#K4" title="NORMSP_1:func.4">-</a> <span class="p3">(<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></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1</span>)</span> <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="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K4" title="NORMSP_1:func.4">-</a> <span class="p3">(<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></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1</span>)</span> <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="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<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#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D6" title="NORMSP_1:def.6">NORMSP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1</span>)</span> <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> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">X</font>)
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1</span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:11_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1</span>)</span> <a href="bhsp_1.html#K8" title="BHSP_1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<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#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<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></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:11_2"/>
<font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1 <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1</span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<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>
 <b>by </b><i><a class="txt" href="bhsp_4.html#E2:11_2"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="bhsp_1.html#T83" title="BHSP_1:th.83">BHSP_1:83</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:11"/><b>then </b><i><font color="Green" title="E11">A6</font></i>: 
<font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1 is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a>
 
<b>by </b><i><a class="txt" href="bhsp_4.html#E2:11"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="bhsp_4.html#E3:11"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="bhsp_2.html#T4" title="BHSP_2:th.4">BHSP_2:4</a></i>;<br/>
<b>hence </b><a NAME="E7:11"/>
<font color="Maroon" title="c2">seq</font> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a>
 <b>by </b><i><a class="ref" href="bhsp_3.html#T46" title="BHSP_3:th.46">BHSP_3:46</a></i>;<br/>

<a NAME="E8:11"/>
<font color="Maroon" title="c2">seq</font> <a href="bhsp_3.html#K2" title="BHSP_3:func.2">^\</a> 1 is    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c2">seq</font>
 
<b>by </b><i><a class="ref" href="bhsp_3.html#T43" title="BHSP_3:th.43">BHSP_3:43</a></i>;<br/>
<b>hence </b><a NAME="E9:11"/>
 <a href="bhsp_2.html#K1" title="BHSP_2:func.1">lim</a> <font color="Maroon" title="c2">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="bhsp_4.html#E4:11"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="bhsp_4.html#E5:11"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="bhsp_4.html#E6:11"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="bhsp_3.html#T33" title="BHSP_3:th.33">BHSP_3:33</a>, <a class="ref" href="bhsp_3.html#T46" title="BHSP_3:th.46">BHSP_3:46</a></i>;<br/>


</div>
