<?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="E1:22"/><i><font color="Green">E29</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 ;<br/>

<b>let </b><font color="Maroon">lp</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="normsp_1.html#L1">NORMSTR</a> ;<br/>

<b>assume </b><a NAME="E2:22"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">lp</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#G1">NORMSTR</a>(# <span class="p1">(<span class="default"><a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K10">Zero_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">p</font></span>)</span>,<a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">p</font></span>)</span>,<a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" target="_self">the_set_of_RealSequences_l^</a> <font color="Maroon">p</font></span>)</span>,<a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="lp_space.html#K3" target="_self">l_norm^</a> <font color="Maroon">p</font></span>)</span> #)
 ;<br/>

<a NAME="E3:22"/><i><font color="Green">E35</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:22"/><i><font color="Green">E36</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">lp</font>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">lp</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<a NAME="E5:22"/>
( 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="E6:22"/><b>then </b>
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E7:22"/><b>then </b><i><font color="Green">E38</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/>
<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/><i><font color="Green">E56</font></i>: <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> = 
the <a href="normsp_1.html#U1">norm</a> of <font color="Maroon">lp</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
<br/>.= 
<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"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</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>
<b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self">Def1</a>, </i>
;<br/><a NAME="E3:22_1"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</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="lp_space.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E4:22_1"/><i><font color="Green">E58</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  0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 <b>by </b><i>, <a class="txt" href="lp_space.html#E2:22"><i><font color="Green">E31</font></i></a>, , , </i>;<br/><a NAME="E5:22_1"/>
<font color="Maroon">x</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="lp_space.html#D1" target="_self">Def1</a>, </i>;<br/><b>hence </b><font color="Maroon">x</font> = 
 <a href="rsspace.html#K6">Zeroseq</a> 
<b>by </b><i>, <a class="ref" href="rsspace.html#D6">RSSPACE:def 6</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">lp</font>
<b>by </b><i>, <a class="ref" href="lp_space.html#D1" target="_self">Def1</a>, </i>
;<br/><br/></div><b>end;</b></div>
<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">lp</font>
 ;<br/><a NAME="E2:22_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rsspace.html#K6">Zeroseq</a> 
 <b>by </b><i>, <a class="ref" href="lp_space.html#D1" target="_self">Def1</a>, , </i>;<br/><a NAME="E3:22_2"/><b>then </b><i><font color="Green">E61</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="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rsspace.html#D6">RSSPACE:def 6</a></i>;<br/><b>thus </b><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> = 
the <a href="normsp_1.html#U1">norm</a> of <font color="Maroon">lp</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
<br/>.= 
<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"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</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>
<b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self">Def1</a>, </i>
<br/>.= 
0
<b>by </b><i>, , </i>
;<br/></div><b>end;</b></div>
<a NAME="E10:22"/><i><font color="Green">E62</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  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">x</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><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/>

<b>set </b><font color="Maroon">xp</font> = <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font></span>)</span> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">p</font>;<br/>
<a NAME="E1:22_3"/><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">x</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">x</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 NAME="E2:22_3"/><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">x</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/>
<a NAME="E3:22_3"/><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">x</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">x</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="E4:22_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">x</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#E3:22"><i><font color="Green">E35</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="E11:22"/>
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</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="lp_space.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E12:22"/><b>then </b><i><font color="Green">E66</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</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="series_1.html#T21">SERIES_1:21</a></i>;<br/>
<a NAME="E13:22"/><i><font color="Green">E73</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"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</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#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="lp_space.html#E1:22"><i><font color="Green">E29</font></i></a>, , </i>;<br/>
<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"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</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> = 
the <a href="normsp_1.html#U1">norm</a> of <font color="Maroon">lp</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self">Def1</a>, </i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
;<br/>
<a NAME="E15:22"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_5"><b>now </b></a><div class="add"><i><font color="Green">E76</font></i>: <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> = 
the <a href="normsp_1.html#U1">norm</a> of <font color="Maroon">lp</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
<br/>.= 
<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"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</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>
<b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self">Def1</a>, </i>
;<br/><i><font color="Green">E77</font></i>: <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">y</font></span><a href="normsp_1.html#K1">.||</a></span> = 
the <a href="normsp_1.html#U1">norm</a> of <font color="Maroon">lp</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
<br/>.= 
<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"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">y</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>
<b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self">Def1</a>, </i>
;<br/><i><font color="Green">E78</font></i>: <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</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">y</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">x</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">y</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">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="lp_space.html#D1" target="_self">Def1</a>, </i>
;<br/><i><font color="Green">E79</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"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</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">y</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="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> = 
the <a href="normsp_1.html#U1">norm</a> of <font color="Maroon">lp</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self">Def1</a>, <a class="ref" href="lp_space.html#D2" target="_self">Def2</a>, </i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
;<br/><a NAME="E5:22_5"/><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</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="lp_space.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E6:22_5"/>
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">y</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="lp_space.html#D1" target="_self">Def1</a>, </i>;<br/><b>hence </b><a NAME="E7:22_5"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</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">x</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">y</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, , <a class="txt" href="lp_space.html#E7:22"><i><font color="Green">E38</font></i></a>, , , </i>;<br/></div><b>end;</b></div>
<i><font color="Green">E81</font></i>: <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> = 
the <a href="normsp_1.html#U1">norm</a> of <font color="Maroon">lp</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
<br/>.= 
<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"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</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>
<b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self">Def1</a>, </i>
;<br/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> = 
the <a href="normsp_1.html#U1">norm</a> of <font color="Maroon">lp</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
<br/>.= 
<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"><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">x</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="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="ref" href="lp_space.html#D1" target="_self">Def1</a>, </i>
<br/>.= 
<span class="p1">(<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="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span>
<b>by </b><i>, <a class="ref" href="lp_space.html#D1" target="_self">Def1</a>, , </i>
;<br/>
<b>hence </b><a NAME="E19:22"/>
( ( <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">lp</font> ) &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">lp</font> implies <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a> 0 ) &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</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">x</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">y</font></span><a href="normsp_1.html#K1">.||</a></span> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <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="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x</font></span><a href="normsp_1.html#K1">.||</a></span> )
 <b>by </b><i>, <a class="txt" href="lp_space.html#E2:22"><i><font color="Green">E31</font></i></a>, <a class="txt" href="lp_space.html#E3:22"><i><font color="Green">E35</font></i></a>, <a class="txt" href="lp_space.html#E4:22"><i><font color="Green">E36</font></i></a></i>;<br/>


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