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

<b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:10"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<b>set </b><font color="Maroon">W</font> =  <a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">p</font>;<br/>
<a NAME="E2:10"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">v</font>, <font color="Olive">u</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">p</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">p</font> holds <br/><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> , <font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> ;<br/>

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">p</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">p</font> )
 ;<br/>

<a NAME="E2:10_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> is <a href="series_1.html#V1">summable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E2:10_1_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:10_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E4:10_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E5:10_1_1"/><i><font color="Green">E57</font></i>: 
1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<b>set </b><font color="Maroon">p1</font> = 1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">vp</font> = <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">up</font> = <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font>;<br/>
<i><font color="Green">E58</font></i>: <font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
;<br/>
<a NAME="E7:10_1_1"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i/>;<br/>
<a NAME="E8:10_1_1"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><a NAME="E1:10_1_1_2"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E2:10_1_1_2"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E3:10_1_1_2"/><i><font color="Green">E64</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>thus </b><a NAME="E4:10_1_1_2"/><i><font color="Green">E65</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> or 0 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self">Def1</a></i>;<br/><b>thus </b><a NAME="E5:10_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="power.html#T39">POWER:39</a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><b>thus </b><a NAME="E6:10_1_1_2"/><i><font color="Green">E66</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>thus </b><a NAME="E7:10_1_1_2"/><i><font color="Green">E73</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> or 0 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E8:10_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="txt" href="lp_space.html#E1:10"><i><font color="Green">E29</font></i></a>, <a class="ref" href="power.html#T39">POWER:39</a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:10_1_1"/><i><font color="Green">E74</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> is <a href="seqm_3.html#V3">non-decreasing</a>
 
<b>by </b><i>, <a class="ref" href="series_1.html#T19">SERIES_1:19</a></i>;<br/>
<a NAME="E11:10_1_1"/><i><font color="Green">E75</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> is <a href="seq_2.html#V1">bounded_above</a>
 
<b>by </b><i>, , <a class="ref" href="series_1.html#T20">SERIES_1:20</a></i>;<br/>
<b>consider </b><font color="Maroon">r1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E13:10_1_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i>, <a class="ref" href="seq_2.html#D1">SEQ_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">r1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E15:10_1_1"/><i><font color="Green">E77</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#T4">SEQ_1:4</a></i>;<br/>
<div><i><font color="Green">E78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:10_1_1_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:10_1_1_3"/><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self">Def1</a>, <a class="txt" href="lp_space.html#E1:10"><i><font color="Green">E29</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="txt" href="lp_space.html#E1:10"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E4:10_1_1_3"/><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i/>;<br/><a NAME="E5:10_1_1_3"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="seqm_3.html#T21">SEQM_3:21</a></i>;<br/><a NAME="E6:10_1_1_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> 0
 <b>by </b><i/>;<br/><a NAME="E7:10_1_1_3"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0
 <b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/><b>hence </b><a NAME="E8:10_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="lp_space.html#E2:10"><i><font color="Green">E31</font></i></a>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E17:10_1_1"/><i><font color="Green">E83</font></i>: 
 ex <font color="Olive">q1</font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">q1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E18:10_1_1"/><i><font color="Green">E84</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> is <a href="seqm_3.html#V3">non-decreasing</a>
 
<b>by </b><i>, <a class="ref" href="series_1.html#T19">SERIES_1:19</a></i>;<br/>
<a NAME="E19:10_1_1"/><i><font color="Green">E95</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> is <a href="seq_2.html#V1">bounded_above</a>
 
<b>by </b><i>, , <a class="ref" href="series_1.html#T20">SERIES_1:20</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E21:10_1_1"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="lp_space.html#E2:10"><i><font color="Green">E31</font></i></a>, <a class="ref" href="seq_2.html#D1">SEQ_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E23:10_1_1"/><i><font color="Green">E98</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#T4">SEQ_1:4</a></i>;<br/>
<div><i><font color="Green">E99</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:10_1_1_4"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:10_1_1_4"/><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="txt" href="lp_space.html#E1:10_1"><i><font color="Green">E35</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="txt" href="lp_space.html#E1:10_1"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E4:10_1_1_4"/><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="lp_space.html#E2:10_1"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E5:10_1_1_4"/><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="seqm_3.html#T21">SEQM_3:21</a></i>;<br/><a NAME="E6:10_1_1_4"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> 0
 <b>by </b><i/>;<br/><a NAME="E7:10_1_1_4"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0
 <b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/><b>hence </b><a NAME="E8:10_1_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="lp_space.html#E1:10_1_1"><i><font color="Green">E38</font></i></a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E25:10_1_1"/><i><font color="Green">E104</font></i>: 
 ex <font color="Olive">q</font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">q</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E27:10_1_1"/><i><font color="Green">E106</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="lp_space.html#E1:10_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">q1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E29:10_1_1"/><i><font color="Green">E107</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q1</font>
 <b>by </b><i><a class="txt" href="lp_space.html#E1:10"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">q</font> <a href="real_1.html#K3">+</a> <font color="Maroon">q1</font>;<br/>
<a NAME="E30:10_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="rsspace.html#K1">the_set_of_RealSequences</a> 
 
<b>by </b><i>, , </i>;<br/>
<b>then reconsider </b><font color="Maroon">vq</font> = <font color="Maroon">v</font> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="ref" href="rsspace.html#D1">RSSPACE:def 1</a></i>;<br/>
<a NAME="E32:10_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="rsspace.html#K1">the_set_of_RealSequences</a> 
 
<b>by </b><i>, , </i>;<br/>
<b>then reconsider </b><font color="Maroon">uq</font> = <font color="Maroon">u</font> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="ref" href="rsspace.html#D1">RSSPACE:def 1</a></i>;<br/>
<a NAME="E34:10_1_1"/><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">vq</font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>;<br/>
<a NAME="E35:10_1_1"/><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">uq</font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>;<br/>
<a NAME="E36:10_1_1"/><i><font color="Green">E113</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="lp_space.html#E1:10_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="lp_space.html#D2" target="_self">Def2</a>, </i>;<br/>
<div><i><font color="Green">E114</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:10_1_1_5"/>
 <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T4">SEQ_1:4</a></i>;<br/><a NAME="E2:10_1_1_5"/><b>then </b><i><font color="Green">E115</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q1</font>
 <b>by </b><i/>;<br/><a NAME="E3:10_1_1_5"/>
 <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T4">SEQ_1:4</a></i>;<br/><a NAME="E4:10_1_1_5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="lp_space.html#E2:10_1"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E5:10_1_1_5"/><b>then </b><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> <a href="real_1.html#K3">+</a> <font color="Maroon">q1</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E6:10_1_1_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:10_1_1_5"/><b>then </b><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> <a href="real_1.html#K3">+</a> <font color="Maroon">q1</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><i><font color="Green">E118</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_1_5_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T38">POWER:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/><br/></div><b>end;</b></div><div><i><font color="Green">E119</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_1_5_2"/><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> = 
0 <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i>, , <a class="ref" href="power.html#D2">POWER:def 2</a></i>
;<br/><br/></div><b>end;</b></div><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:10_1_1_5_3"/><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E2:10_1_1_5_3"/><i><font color="Green">E123</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> or 0 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>thus </b><a NAME="E3:10_1_1_5_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="power.html#T39">POWER:39</a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/></div>
<b>end;</b></div><a NAME="E11:10_1_1_5"/><b>then </b><i><font color="Green">E124</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> is <a href="seqm_3.html#V3">non-decreasing</a>
 <b>by </b><i><a class="ref" href="series_1.html#T19">SERIES_1:19</a></i>;<br/><div><i><font color="Green">E125</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_5_4"><b>now </b></a><div class="add"><a NAME="E1:10_1_1_5_4"/><i><font color="Green">E126</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="seqm_3.html#T21">SEQM_3:21</a></i>;<br/><a NAME="E2:10_1_1_5_4"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:10_1_1_5_4"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0
 <b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:10_1_1_5_4"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E13:10_1_1_5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E14:10_1_1_5"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="real_1.html#K3">+</a> <font color="Maroon">q1</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="txt" href="lp_space.html#E2:10_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="lp_space.html#E3:10_1_1"><i><font color="Green">E45</font></i></a>, , , </i>;<br/></div><b>end;</b></div>
<a NAME="E38:10_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> (  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> is <a href="seq_2.html#V1">bounded_above</a> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> )
 
<b>by </b><i>, <a class="ref" href="seq_2.html#D3">SEQ_2:def 3</a></i>;<br/>
<a NAME="E39:10_1_1"/><b>then </b><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">vq</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">uq</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="series_1.html#T20">SERIES_1:20</a></i>;<br/>
<a NAME="E40:10_1_1"/><i><font color="Green">E128</font></i>: 
 <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">vq</font>
 
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>;<br/>
<a NAME="E41:10_1_1"/>
 <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">uq</font>
 
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>;<br/>
<a NAME="E42:10_1_1"/><b>then </b><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i>, </i>;<br/>
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span> = 
 <a href="rsspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>
<br/>.= 
 <a href="rsspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T4">RSSPACE:4</a>, <a class="ref" href="rsspace.html#D7">RSSPACE:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E44:10_1_1"/>
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> is <a href="series_1.html#V1">summable</a>
 <b>by </b><i><a class="txt" href="lp_space.html#E2:10_1_1"><i><font color="Green">E44</font></i></a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E130</font></i>: <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> = 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T4">RSSPACE:4</a>, <a class="ref" href="rsspace.html#D7">RSSPACE:def 7</a></i>
<br/>.= 
 <a href="rsspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>
<br/>.= 
 <a href="rsspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T4">RSSPACE:4</a>, <a class="ref" href="rsspace.html#D7">RSSPACE:def 7</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">w</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E6:10_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="rsspace.html#K1">the_set_of_RealSequences</a> 
 
<b>by </b><i><a class="ref" href="rsspace.html#D1">RSSPACE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:10_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">p</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:10"/>
for <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">p</font> holds <br/><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> ;<br/>



<b>assume </b><a NAME="E1:10_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:10_2"/>
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> is <a href="series_1.html#V1">summable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">vp</font> = <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font>;<br/>
<a NAME="E1:10_2_1"/><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E2:10_2_1"/>
( 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i/>;<br/>
<a NAME="E3:10_2_1"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<b>set </b><font color="Maroon">p1</font> = 1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font>;<br/>
<a NAME="E4:10_2_1"/><i><font color="Green">E135</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:10_2_1"/><i><font color="Green">E136</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E137</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:10_2_1_1"/><i><font color="Green">E138</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E2:10_2_1_1"/><i><font color="Green">E139</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>thus </b><a NAME="E3:10_2_1_1"/><i><font color="Green">E140</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> or 0 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:10_2_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="power.html#T39">POWER:39</a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:10_2_1"/><i><font color="Green">E141</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> is <a href="seq_2.html#V1">bounded_above</a>
 
<b>by </b><i>, , <a class="ref" href="series_1.html#T20">SERIES_1:20</a></i>;<br/>
<a NAME="E8:10_2_1"/><i><font color="Green">E142</font></i>: 
 ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#D1">SEQ_2:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">q1</font> = <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font>;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<a NAME="E10:10_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="rsspace.html#K1">the_set_of_RealSequences</a> 
 
<b>by </b><i>, , </i>;<br/>
<b>then reconsider </b><font color="Maroon">vq</font> = <font color="Maroon">v</font> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="ref" href="rsspace.html#D1">RSSPACE:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <a href="rsspace.html#K1">the_set_of_RealSequences</a> ,<a href="rsspace.html#K6">Zeroseq</a> ,<a href="rsspace.html#K4">l_add</a> ,<a href="rsspace.html#K5">l_mult</a>  #) <b>by </b><i>, , </i>;<br/>
<i><font color="Green">E144</font></i>:  <a href="rsspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> = 
 <a href="rsspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T5">RSSPACE:5</a>, <a class="ref" href="rsspace.html#D7">RSSPACE:def 7</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>
;<br/>
<a NAME="E14:10_2_1"/><i><font color="Green">E145</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_3"><b>proof </b></a><div class="add">

<div><i><font color="Green">E146</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:10_2_1_3_1"/><i><font color="Green">E147</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E2:10_2_1_3_1"/><i><font color="Green">E148</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E3:10_2_1_3_1"/><i><font color="Green">E149</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E4:10_2_1_3_1"/><i><font color="Green">E150</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, , , </i>
;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self">Def1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E2:10_2_1_3"/><i><font color="Green">E151</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:10_2_1_3_2"/>
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <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="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:10_2_1_3_2"/>
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T8">SEQ_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:10_2_1_3"/><i><font color="Green">E152</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="series_1.html#T12">SERIES_1:12</a></i>;<br/>
<a NAME="E4:10_2_1_3"/><i><font color="Green">E153</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>thus </b><a NAME="E5:10_2_1_3"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self">Def1</a>, <a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E16:10_2_1"/><i><font color="Green">E154</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>
<a NAME="E17:10_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E18:10_2_1"/><b>then </b><i><font color="Green">E155</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> or 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E156</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:10_2_1_4"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:10_2_1_4"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="seq_1.html#T3">SEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E3:10_2_1_4"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self">Def1</a>, <a class="ref" href="seq_1.html#T3">SEQ_1:3</a></i>;<br/><a NAME="E4:10_2_1_4"/><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="lp_space.html#D1" target="_self">Def1</a></i>;<br/><b>thus </b><a NAME="E5:10_2_1_4"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> )
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/></div><b>end;</b></div>
<a NAME="E20:10_2_1"/><i><font color="Green">E159</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> or <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> or <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:10_2_1_5"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:10_2_1_5"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i>, <a class="ref" href="seq_1.html#T3">SEQ_1:3</a></i>;<br/>
<a NAME="E3:10_2_1_5"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T3">SEQ_1:3</a></i>;<br/>
<a NAME="E4:10_2_1_5"/><b>then </b><i><font color="Green">E162</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> )
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E6:10_2_1_5"/>
( <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> or <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> )
 
<b>by </b><i>, <a class="txt" href="lp_space.html#E1:10"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:10_2_1_5"/>
( <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> or <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E21:10_2_1"/><i><font color="Green">E163</font></i>: 
 ex <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_6"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">r1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1;<br/>

<b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">r1</font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<a NAME="E2:10_2_1_6"/><i><font color="Green">E164</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:10_2_1_6_1"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:10_2_1_6_1"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="lp_space.html#E1:10"><i><font color="Green">E29</font></i></a>, <a class="ref" href="seq_1.html#T3">SEQ_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E4:10_2_1_6_1"/><i><font color="Green">E167</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E5:10_2_1_6_1"/><i><font color="Green">E168</font></i>: 
( <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> implies <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="lp_space.html#E2:10"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:10_2_1_6_1"/>
( <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> implies <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E7:10_2_1_6_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self">Def1</a>, <a class="txt" href="lp_space.html#E1:10"><i><font color="Green">E29</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:10_2_1_6"/>
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E22:10_2_1"/><i><font color="Green">E169</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:10_2_1_7"/><i><font color="Green">E170</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:10_2_1_7"/><i><font color="Green">E171</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>;<br/>
<b>set </b><font color="Maroon">b</font> = <font color="Maroon">a</font> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span>;<br/>
<a NAME="E3:10_2_1_7"/><i><font color="Green">E172</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:10_2_1_7"/><i><font color="Green">E173</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:10_2_1_7"/>
( 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i/>;<br/>
<a NAME="E6:10_2_1_7"/><b>then </b><i><font color="Green">E174</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E7:10_2_1_7"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<b>hence </b><a NAME="E8:10_2_1_7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="lp_space.html#E1:10"><i><font color="Green">E29</font></i></a>, <a class="txt" href="lp_space.html#E2:10"><i><font color="Green">E31</font></i></a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:10_2_1"/>
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span> is <a href="seq_2.html#V1">bounded_above</a>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#D1">SEQ_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E24:10_2_1"/>
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> is <a href="series_1.html#V1">summable</a>
 <b>by </b><i>, <a class="ref" href="series_1.html#T20">SERIES_1:20</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_2"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="rsspace.html#D7">RSSPACE:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:10"/>
 <a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">p</font> is <a href="rlsub_1.html#V1">lineary-closed</a>
 <b>by </b><i>, <a class="ref" href="rlsub_1.html#D1">RLSUB_1:def 1</a></i>;<br/>


</div>
