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

<b>let </b><font color="Maroon">a</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>, <font color="Maroon">b</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:8"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E2:8"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">a</font> <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>and </b><br/><a NAME="E3:8"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">b</font> <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>
 ;<br/>

<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_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/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">b</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 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#K9">+</a> <font color="Maroon">b</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/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">b</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="power.html#T30">POWER:30</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <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"><font color="Maroon">a</font> <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/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">b</font> <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"><font color="Maroon">b</font> <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/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E5:8"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">b</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>  <a href="seq_1.html#K22">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>;<br/>
<a NAME="E6:8"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">a</font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a>  <a href="seq_1.html#K22">abs</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>;<br/>
<a NAME="E7:8"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">b</font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a>  <a href="seq_1.html#K22">abs</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>;<br/>
<a NAME="E8:8"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">b</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="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="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
<a NAME="E9:8"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></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="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
<a NAME="E10:8"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></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="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
<a NAME="E11:8"/><i><font color="Green">E59</font></i>: 
 <a href="seq_1.html#K22">abs</a> <font color="Maroon">a</font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:8"/><i><font color="Green">E60</font></i>: 
 <a href="seq_1.html#K22">abs</a> <font color="Maroon">b</font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:8"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> is <a href="series_1.html#V2">absolutely_summable</a>
 
<b>by </b><i>, <a class="ref" href="series_1.html#D5">SERIES_1:def 5</a></i>;<br/>
<a NAME="E14:8"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">b</font> is <a href="series_1.html#V2">absolutely_summable</a>
 
<b>by </b><i>, <a class="ref" href="series_1.html#D5">SERIES_1:def 5</a></i>;<br/>
<a NAME="E15:8"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">a</font> <a href="funct_2.html#R2">=</a>  <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="funct_2.html#R2">=</a>  <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>;<br/>
<a NAME="E16:8"/><i><font color="Green">E64</font></i>: 
(  <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">a</font> is <a href="series_1.html#V2">absolutely_summable</a> &amp;  <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">b</font> is <a href="series_1.html#V2">absolutely_summable</a> )
 
<b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace3.html#K3">l1_Space</a>  <b>by </b><i>, <a class="ref" href="rsspace3.html#T8">RSSPACE3:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace3.html#K3">l1_Space</a>  <b>by </b><i>, <a class="ref" href="rsspace3.html#T8">RSSPACE3:8</a></i>;<br/>
<a NAME="E19:8"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">a1</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">b1</font></span><a href="normsp_1.html#K1">.||</a></span>
 
<b>by </b><i><a class="ref" href="rsspace3.html#T9">RSSPACE3:9</a></i>;<br/>
<i><font color="Green">E66</font></i>: <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b1</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
the <a href="normsp_1.html#U1">norm</a> of <a href="rsspace3.html#K3">l1_Space</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
<br/>.= 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace3.html#D2">RSSPACE3:def 2</a>, <a class="ref" href="rsspace3.html#D3">RSSPACE3:def 3</a></i>
<br/>.= 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">a1</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace3.html#T8">RSSPACE3:8</a></i>
<br/>.= 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">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">a1</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>
;<br/>
<i><font color="Green">E73</font></i>: <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">a1</font></span><a href="normsp_1.html#K1">.||</a></span> = 
the <a href="normsp_1.html#U1">norm</a> of <a href="rsspace3.html#K3">l1_Space</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">a1</font>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
<br/>.= 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">a1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace3.html#D2">RSSPACE3:def 2</a>, <a class="ref" href="rsspace3.html#D3">RSSPACE3:def 3</a></i>
;<br/>
<i><font color="Green">E74</font></i>: <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">b1</font></span><a href="normsp_1.html#K1">.||</a></span> = 
the <a href="normsp_1.html#U1">norm</a> of <a href="rsspace3.html#K3">l1_Space</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
<br/>.= 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace3.html#D2">RSSPACE3:def 2</a>, <a class="ref" href="rsspace3.html#D3">RSSPACE3:def 3</a></i>
;<br/>
<a NAME="E23:8"/><i><font color="Green">E75</font></i>: 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">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">a1</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">a1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="lp_space.html#D1" target="_self">Def1</a>, , </i>;<br/>
<a NAME="E24:8"/><i><font color="Green">E76</font></i>: 
 <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">a1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>;<br/>
<a NAME="E25:8"/><i><font color="Green">E77</font></i>: 
 <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">b1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>;<br/>
<a NAME="E26:8"/><i><font color="Green">E78</font></i>: 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="lp_space.html#E1:8"><i><font color="Green">E29</font></i></a>, , <a class="txt" href="lp_space.html#E2:8"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E27:8"/><i><font color="Green">E79</font></i>: 
( <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">b</font></span>)</span> is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">b</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="series_1.html#T10">SERIES_1:10</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</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="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E29:8"/><i><font color="Green">E80</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  <font color="Maroon">c</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<div><i><font color="Green">E81</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_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="E1:8_5"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <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">a</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">b</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#T16">SEQ_1:16</a></i>;<br/><b>hence </b><a NAME="E2:8_5"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">b</font></span>)</span></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="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E3:8_5"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <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">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/><a NAME="E4:8_5"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">b</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"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lp_space.html#E4:8"><i><font color="Green">E36</font></i></a>, <a class="ref" href="complex1.html#T142">COMPLEX1:142</a></i>;<br/><b>thus </b><a NAME="E5:8_5"/>
<font color="Maroon">c</font> <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="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><i><font color="Green">E84</font></i>: <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="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">b</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#T16">SEQ_1:16</a></i>
;<br/><b>thus </b><a NAME="E7:8_5"/>
<font color="Maroon">c</font> <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="seq_1.html#K22">abs</a> <font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="lp_space.html#E5:8"><i><font color="Green">E38</font></i></a></i>;<br/><b>thus </b><a NAME="E8:8_5"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">b</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">c</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E31:8"/>
<font color="Maroon">c</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">a</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lp_space.html#E3:8"><i><font color="Green">E35</font></i></a>, <a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>
<b>hence </b><a NAME="E32:8"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">b</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="txt" href="lp_space.html#E3:8"><i><font color="Green">E35</font></i></a>, <a class="ref" href="series_1.html#T24">SERIES_1:24</a></i>;<br/>

<b>thus </b><a NAME="E33:8"/>
<span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">b</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="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"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></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"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font></span>)</span></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>, , , </i>;<br/>


</div>
