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

<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E20">A1</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">lp</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="normsp_1.html#L1" title="NORMSP_1:struct.1">NORMSTR</a> ;<br/>

<b>assume </b><a NAME="E2:19"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c2">lp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#G1" title="NORMSP_1:aggr.1">NORMSTR</a>(# <span class="p1">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K10" title="RSSPACE:func.10">Zero_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K8" title="RSSPACE:func.8">Add_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K9" title="RSSPACE:func.9">Mult_</a> <span class="p2">(<span class="default"><a href="lp_space.html#K2" title="LP_SPACE:func.2">the_set_of_RealSequences_l^</a> <font color="Maroon" title="c1">p</font></span>)</span>,<a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span>)</span>,<span class="p1">(<span class="default"><a href="lp_space.html#K3" title="LP_SPACE:func.3">l_norm^</a> <font color="Maroon" title="c1">p</font></span>)</span> #)
 ;<br/>

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">lp</font>;<br/><b>let </b><font color="Maroon" title="c4">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">n1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:19_1"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c5">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><b>reconsider </b><font color="Maroon" title="c6">n</font> = <font color="Maroon" title="c5">n1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="lp_space.html#E1:19_1"><i><font color="Green" title="E22">A3</font></i></a></i>;<br/><a NAME="E3:19_1"/><i><font color="Green" title="E23">A4</font></i>: 
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/><i><font color="Green" title="E24">A5</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self" title="LP_SPACE:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p5">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="txt" href="lp_space.html#E1:19"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="lp_space.html#E2:19"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="lp_space.html#T10" target="_self" title="LP_SPACE:th.10">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="ref" href="rsspace.html#T3" title="RSSPACE:th.3">RSSPACE:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span>
<b>by </b><i><a class="txt" href="lp_space.html#E1:19"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="lp_space.html#E3:19_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="lp_space.html#E4"><i><font color="Green" title="E4">Lm2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self" title="LP_SPACE:def.1">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" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c7">n</font> = <font color="Maroon" title="c6">n</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>thus </b><a NAME="E6:19_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n1</font>
 <b>by </b><i><a class="txt" href="lp_space.html#E4:19_1"><i><font color="Green" title="E24">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:19"/>
<span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="lp_space.html#K1" title="LP_SPACE:func.1">rto_power</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T8" title="SEQ_1:th.8">SEQ_1:8</a></i>;<br/>


</div>
