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

<a NAME="E1:1"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  holds  <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">v</font></span>)</span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c1">q0</font> = 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">p</font> = <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">q</font> = <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">r</font> =  <a href="valued_1.html#K56" title="VALUED_1:func.56">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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:1_1"/><span class="lab"><font color="Green" title="E3">A2</font></span>: 
for <font color="Olive" title="b1">n</font> being   <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>  holds   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c7">rr</font> = <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">n</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">tt</font> =  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E2:1_1_1"/><span class="lab"><font color="Green" title="E4">A3</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">tt</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></span>;<br/>
<span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font> = 
2 <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="valued_1.html#K56" title="VALUED_1:func.56">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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <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"><span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T16" title="SEQ_1:th.16">SEQ_1:16</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:1_1"/><span class="lab"><font color="Green" title="E4">A4</font></span>: 
for <font color="Olive" title="b1">n</font> being   <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>  holds  <span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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">v</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c7">s</font> =  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">t</font> =  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">n</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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">v</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">sn</font> = <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font>, <font color="Maroon" title="c11">tn</font> = <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">ss</font> =  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c10">sn</font>, <font color="Maroon" title="c13">tt</font> =  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c11">tn</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="lab"><font color="Green" title="E5">A5</font></span>: <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">v</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font> = 
<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">v</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></span>
<br/>.= 
<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">v</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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">v</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">sn</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c2">u</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c2">u</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></span>
<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="c10">sn</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">tn</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T161" title="COMPLEX1:th.161">COMPLEX1:161</a></span>
<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="c10">sn</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c11">tn</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T161" title="COMPLEX1:th.161">COMPLEX1:161</a></span>
;<br/>
<a NAME="E4:1_1_2"/><span class="lab"><font color="Green" title="E6">A6</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c10">sn</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c11">tn</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></span>;<br/>
<span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font> = 
2 <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="valued_1.html#K56" title="VALUED_1:func.56">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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></span>
<br/>.= 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <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"><span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T16" title="SEQ_1:th.16">SEQ_1:16</a></span>
<br/>.= 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <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"><span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <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">v</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></span>
<br/>.= 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">ss</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">tt</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <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> <font color="Maroon" title="c10">sn</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c11">tn</font></span>)</span>

;<br/>
<a NAME="E6:1_1_2"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p0">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_2"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="txt" href="#E4:1_1_2"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:1_1_2"/>
<span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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">v</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E5">A7</font></span>: <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">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> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T31" title="SEQ_1:th.31">SEQ_1:31</a></span>
<br/>.= 
 <a href="valued_1.html#K56" title="VALUED_1:func.56">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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T35" title="SEQ_1:th.35">SEQ_1:35</a></span>
;<br/>
<a NAME="E4:1_1"/>
( <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  &amp; <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#D11" title="RSSPACE:def.11">RSSPACE:def 11</a>, <a class="ref" href="rsspace.html#D13" title="RSSPACE:def.13">RSSPACE:def 13</a></span>;<br/>
<a NAME="E5:1_1"/><span class="kw">then </span>
<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">v</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span></span>)</span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T10" title="SERIES_1:th.10">SERIES_1:10</a></span>;<br/>
<a NAME="E6:1_1"/><span class="kw">then </span>
2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">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> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span></span>)</span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="ref" href="series_1.html#T24" title="SERIES_1:th.24">SERIES_1:24</a></span>;<br/>
<a NAME="E7:1_1"/><span class="kw">then </span>
 <a href="valued_1.html#K56" title="VALUED_1:func.56">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="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1"><span class="lab"><font color="Green" title="E5">A7</font></span></a>, <a class="ref" href="series_1.html#T13" title="SERIES_1:th.13">SERIES_1:13</a></span>;<br/>
<a NAME="E8:1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span> is  <a href="series_1.html#V2" title="SERIES_1:attr.2">absolutely_summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D5" title="SERIES_1:def.5">SERIES_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:1_1"/>
<span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T40" title="SERIES_1:th.40">SERIES_1:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:1"/><span class="lab"><font color="Green" title="E3">A8</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  iff ( <font color="Olive" title="b1">x</font> is   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> &amp; <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">x</font></span>)</span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  iff ( <font color="Maroon" title="c1">x</font> is   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> &amp; <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  ) )</span><br/>

<a NAME="E1:1_2"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace.html#K1" title="RSSPACE:func.1">the_set_of_RealSequences</a>  iff <font color="Maroon" title="c1">x</font> is   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#D1" title="RSSPACE:def.1">RSSPACE:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:1_2"/>
( <font color="Maroon" title="c1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  iff ( <font color="Maroon" title="c1">x</font> is   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> &amp; <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#D11" title="RSSPACE:def.11">RSSPACE:def 11</a>, <a class="ref" href="rsspace.html#D13" title="RSSPACE:def.13">RSSPACE:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:1"/><span class="lab"><font color="Green" title="E4">A9</font></span>: 
for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  holds  <font color="Olive" title="b1">v</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</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="Olive" title="b1">v</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">w</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">v</font>, <font color="Maroon" title="c2">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">v</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</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="c1">v</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">w</font></span>)</span></span>)</span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c1">v</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c2">w</font> = 
the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">v</font>,<font color="Maroon" title="c2">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_1.html#D1" title="BHSP_1:def.1">BHSP_1:def 1</a></span>
<br/>.= 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</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="c1">v</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">w</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#D12" title="RSSPACE:def.12">RSSPACE:def 12</a>, <a class="ref" href="rsspace.html#D13" title="RSSPACE:def.13">RSSPACE:def 13</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:1"/><span class="lab"><font color="Green" title="E5">A10</font></span>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  holds  <font color="Olive" title="b1">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">v</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c1">W</font> =  <a href="rsspace.html#K11" title="RSSPACE:func.11">the_set_of_l2RealSequences</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">L1</font> =  <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c4">v</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">u1</font> = <font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c4">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_1.html#T18" title="RLSUB_1:th.18">RLSUB_1:18</a>, <a class="ref" href="rsspace.html#T15" title="RSSPACE:th.15">RSSPACE:15</a>, <a class="ref" href="rsspace.html#D13" title="RSSPACE:def.13">RSSPACE:def 13</a></span>;<br/>
<a NAME="E2:1_4"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:1_4"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="rsspace.html#K11" title="RSSPACE:func.11">the_set_of_l2RealSequences</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="rsspace.html#K11" title="RSSPACE:func.11">the_set_of_l2RealSequences</a> ,<a href="rsspace.html#K11" title="RSSPACE:func.11">the_set_of_l2RealSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></span>;<br/>
<a NAME="E4:1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A11</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="rsspace.html#K11" title="RSSPACE:func.11">the_set_of_l2RealSequences</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#D13" title="RSSPACE:def.13">RSSPACE:def 13</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">v</font> = 
<span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="rsspace.html#K11" title="RSSPACE:func.11">the_set_of_l2RealSequences</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#D8" title="RSSPACE:def.8">RSSPACE:def 8</a>, <a class="ref" href="rsspace.html#D13" title="RSSPACE:def.13">RSSPACE:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c5">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">v1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_4"><span class="lab"><font color="Green" title="E6">A11</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:1_4"/>
<font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c4">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#T17" title="RSSPACE:th.17">RSSPACE:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:1"/><span class="lab"><font color="Green" title="E6">A12</font></span>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  holds  <font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c1">W</font> =  <a href="rsspace.html#K11" title="RSSPACE:func.11">the_set_of_l2RealSequences</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">L1</font> =  <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  holds  <font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c4">u</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c5">u1</font> = <font color="Maroon" title="c4">u</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_1.html#T18" title="RLSUB_1:th.18">RLSUB_1:18</a>, <a class="ref" href="rsspace.html#T15" title="RSSPACE:th.15">RSSPACE:15</a>, <a class="ref" href="rsspace.html#D13" title="RSSPACE:def.13">RSSPACE:def 13</a></span>;<br/>
<a NAME="E2:1_5"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:1_5"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="rsspace.html#K11" title="RSSPACE:func.11">the_set_of_l2RealSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="rsspace.html#K11" title="RSSPACE:func.11">the_set_of_l2RealSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></span>;<br/>
<a NAME="E4:1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A13</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="rsspace.html#K11" title="RSSPACE:func.11">the_set_of_l2RealSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#D13" title="RSSPACE:def.13">RSSPACE:def 13</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font> = 
<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="rsspace.html#K11" title="RSSPACE:func.11">the_set_of_l2RealSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#D9" title="RSSPACE:def.9">RSSPACE:def 9</a>, <a class="ref" href="rsspace.html#D13" title="RSSPACE:def.13">RSSPACE:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_5"><span class="lab"><font color="Green" title="E7">A13</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:1_5"/>
<font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c4">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#T17" title="RSSPACE:th.17">RSSPACE:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:1"/><span class="lab"><font color="Green" title="E7">A14</font></span>: 
for <font color="Olive" title="b1">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  holds  <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span><br/>

<a NAME="E1:1_6"/>
<font color="Maroon" title="c1">u</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_1.html#T18" title="RLSUB_1:th.18">RLSUB_1:18</a>, <a class="ref" href="rsspace.html#T15" title="RSSPACE:th.15">RSSPACE:15</a>, <a class="ref" href="rsspace.html#D13" title="RSSPACE:def.13">RSSPACE:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:1_6"/>
<font color="Maroon" title="c1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#T17" title="RSSPACE:th.17">RSSPACE:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:1"/><span class="lab"><font color="Green" title="E8">A15</font></span>: 
for <font color="Olive" title="b1">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  holds <br/> (  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> &amp;  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span> &amp;  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c1">u</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span> )</span><br/>

 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c1">u</font> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c1">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T29" title="RLVECT_1:th.29">RLVECT_1:29</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1"><span class="lab"><font color="Green" title="E6">A12</font></span></a></span>
<br/>.= 
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T25" title="SEQ_1:th.25">SEQ_1:25</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:1_7"/>
(  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span> &amp;  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c1">u</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1"><span class="lab"><font color="Green" title="E7">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:1"/><span class="lab"><font color="Green" title="E9">A16</font></span>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  holds  <font color="Olive" title="b1">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">v</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">u</font>, <font color="Maroon" title="c2">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c1">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">v</font> = 
<span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1"><span class="lab"><font color="Green" title="E5">A10</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1"><span class="lab"><font color="Green" title="E8">A15</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T15" title="SEQ_1:th.15">SEQ_1:15</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#D10" title="RSSPACE:def.10">RSSPACE:def 10</a>, <a class="ref" href="rsspace.html#D13" title="RSSPACE:def.13">RSSPACE:def 13</a></span>
<br/>.= 
 <a href="rsspace.html#K6" title="RSSPACE:func.6">Zeroseq</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rsspace.html#D7" title="RSSPACE:def.7">RSSPACE:def 7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rsspace.html#K11" title="RSSPACE:func.11">the_set_of_l2RealSequences</a>  &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  iff ( <font color="Olive" title="b1">x</font> is   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> &amp; <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">x</font></span>)</span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  ) ) ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rsspace.html#K6" title="RSSPACE:func.6">Zeroseq</a>  &amp; ( for <font color="Olive" title="b1">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  holds  <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font> ) &amp; ( for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  holds  <font color="Olive" title="b1">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">v</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  holds  <font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">u</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  holds <br/> (  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> &amp;  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> ) ) &amp; ( for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  holds  <font color="Olive" title="b1">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">v</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  holds <br/> ( <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">w</font></span>)</span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  &amp; ( for <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K13" title="RSSPACE:func.13">l2_Space</a>  holds  <font color="Olive" title="b3">v</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</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="Olive" title="b3">v</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b4">w</font></span>)</span></span>)</span> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1"><span class="lab"><font color="Green" title="E3">A8</font></span></a>, <a class="txt" href="#E6:1"><span class="lab"><font color="Green" title="E7">A14</font></span></a>, <a class="txt" href="#E4:1"><span class="lab"><font color="Green" title="E5">A10</font></span></a>, <a class="txt" href="#E5:1"><span class="lab"><font color="Green" title="E6">A12</font></span></a>, <a class="txt" href="#E7:1"><span class="lab"><font color="Green" title="E8">A15</font></span></a>, <a class="txt" href="#E8:1"><span class="lab"><font color="Green" title="E9">A16</font></span></a>, <a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E3:1"><span class="lab"><font color="Green" title="E4">A9</font></span></a>, <a class="ref" href="rsspace.html#D13" title="RSSPACE:def.13">RSSPACE:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
