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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>







<a NAME="E1:56"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</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" target="_self">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="bhsp_4.html#T54" target="_self">Th54</a></i>;<br/>
<a NAME="E2:56"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>;<br/>
<a NAME="E3:56"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>;<br/>
<a NAME="E4:56"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>;<br/>
<a NAME="E5:56"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K9">+</a> <span class="p5">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T15">SEQ_1:15</a></i>;<br/>
<a NAME="E6:56"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><a href="seq_1.html#K17">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T37">SEQ_1:37</a></i>;<br/>
<a NAME="E7:56"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><a href="seq_1.html#K17">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T25">SEQ_1:25</a></i>;<br/>
<a NAME="E8:56"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T25">SEQ_1:25</a></i>;<br/>
<a NAME="E9:56"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T32">SEQ_1:32</a></i>;<br/>
<a NAME="E10:56"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="normsp_1.html#K6">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="bhsp_4.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E11:56"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="normsp_1.html#K6">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="bhsp_4.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E12:56"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D8">NORMSP_1:def 8</a></i>;<br/>
<a NAME="E13:56"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p5">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T29">RLVECT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E14:56"/>
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="bhsp_4.html#K5" target="_self">*</a> <span class="p4">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>


</div>
