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

<a NAME="E1:1"/><i><font color="Green">E2</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <a href="rsspace.html#K13">l2_Space</a>  iff ( <font color="Olive">b<sub>1</sub></font> is   <a href="seq_1.html#NM1">Real_Sequence</a> &amp; <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> is <a href="series_1.html#V1">summable</a> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="rsspace.html#K1">the_set_of_RealSequences</a>  iff <font color="Maroon">c<sub>1</sub></font> is   <a href="seq_1.html#NM1">Real_Sequence</a> )
 
<b>by </b><i><a class="ref" href="rsspace.html#D1">RSSPACE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:1_1"/>
( <font color="Maroon">c<sub>1</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <a href="rsspace.html#K13">l2_Space</a>  iff ( <font color="Maroon">c<sub>1</sub></font> is   <a href="seq_1.html#NM1">Real_Sequence</a> &amp; <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="series_1.html#V1">summable</a> ) )
 <b>by </b><i><a class="ref" href="rsspace.html#D11">RSSPACE:def 11</a>, <a class="ref" href="rsspace.html#D13">RSSPACE:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:1"/><i><font color="Green">E3</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a> ;<br/>

<a NAME="E1:1_2"/>
<font color="Maroon">c<sub>1</sub></font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> 
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T18">RLSUB_1:18</a>, <a class="ref" href="rsspace.html#T15">RSSPACE:15</a>, <a class="ref" href="rsspace.html#D13">RSSPACE:def 13</a></i>;<br/>
<b>hence </b><a NAME="E2:1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="rsspace.html#T17">RSSPACE:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:1"/><i><font color="Green">E4</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a> ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>  <b>by </b><i><a class="ref" href="rlsub_1.html#T18">RLSUB_1:18</a>, <a class="ref" href="rsspace.html#T15">RSSPACE:15</a>, <a class="ref" href="rsspace.html#D13">RSSPACE:def 13</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="rsspace.html#K11">the_set_of_l2RealSequences</a> ;<br/>
<a NAME="E2:1_3"/><i><font color="Green">E5</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="rsspace.html#K11">the_set_of_l2RealSequences</a> ,<a href="rsspace.html#K11">the_set_of_l2RealSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E3:1_3"/>
 <a href="relat_1.html#K1">dom</a> the <a href="rlvect_1.html#U1">add</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:1_3"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>  <a href="realset1.html#K1">||</a> <a href="rsspace.html#K11">the_set_of_l2RealSequences</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="rsspace.html#K11">the_set_of_l2RealSequences</a> ,<a href="rsspace.html#K11">the_set_of_l2RealSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="rsspace2.html#E2:1_3"><i><font color="Green">E5</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E5:1_3"/><b>then </b><i><font color="Green">E6</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>  <a href="realset1.html#K1">||</a> <a href="rsspace.html#K11">the_set_of_l2RealSequences</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="rsspace.html#D13">RSSPACE:def 13</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>2</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <a href="rsspace.html#K13">l2_Space</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>  <a href="realset1.html#K1">||</a> <a href="rsspace.html#K11">the_set_of_l2RealSequences</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="rsspace.html#D8">RSSPACE:def 8</a>, <a class="ref" href="rsspace.html#D13">RSSPACE:def 13</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="rsspace2.html#E5:1_3"><i><font color="Green">E6</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E7:1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rsspace.html#T17">RSSPACE:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:1"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a> ;<br/>



<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>  <b>by </b><i><a class="ref" href="rlsub_1.html#T18">RLSUB_1:18</a>, <a class="ref" href="rsspace.html#T15">RSSPACE:15</a>, <a class="ref" href="rsspace.html#D13">RSSPACE:def 13</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="rsspace.html#K11">the_set_of_l2RealSequences</a> ;<br/>
<a NAME="E2:1_4"/><i><font color="Green">E6</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="rsspace.html#K11">the_set_of_l2RealSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E3:1_4"/>
 <a href="relat_1.html#K1">dom</a> the <a href="rlvect_1.html#U2">Mult</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:1_4"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="rsspace.html#K11">the_set_of_l2RealSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="rsspace.html#K11">the_set_of_l2RealSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="rsspace2.html#E2:1_4"><i><font color="Green">E6</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E5:1_4"/><b>then </b><i><font color="Green">E7</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="rsspace.html#K11">the_set_of_l2RealSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="rsspace.html#D13">RSSPACE:def 13</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> = 
the <a href="rlvect_1.html#U2">Mult</a> of <a href="rsspace.html#K13">l2_Space</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="rsspace.html#K11">the_set_of_l2RealSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="rsspace.html#D9">RSSPACE:def 9</a>, <a class="ref" href="rsspace.html#D13">RSSPACE:def 13</a></i>
<br/>.= 
the <a href="rlvect_1.html#U2">Mult</a> of <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="rsspace2.html#E5:1_4"><i><font color="Green">E7</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E7:1_4"/>
<font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rsspace.html#T17">RSSPACE:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:1"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a>  holds <br/>(  <a href="rlvect_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_1.html#K17">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp;  <a href="rsspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="seq_1.html#K17">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a> ;<br/>

 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T29">RLVECT_1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rsspace2.html#E4:1"><i><font color="Green">E5</font></i></a></i>
<br/>.= 
 <a href="seq_1.html#K17">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T25">SEQ_1:25</a></i>
;<br/>
<b>hence </b><a NAME="E2:1_5"/>
(  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_1.html#K17">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="rsspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="seq_1.html#K17">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="rsspace2.html#E2:1"><i><font color="Green">E3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:1"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a> ;<br/>

<b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rsspace2.html#E3:1"><i><font color="Green">E4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rsspace2.html#E5:1"><i><font color="Green">E6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T15">SEQ_1:15</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E7:1"/><i><font color="Green">E8</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <a href="rsspace.html#K13">l2_Space</a>  <a href="hidden.html#R1">=</a>  <a href="rsspace.html#K6">Zeroseq</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_7"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="rsspace.html#K11">the_set_of_l2RealSequences</a> ;<br/>
<b>thus </b> <a href="rlvect_1.html#K1">0.</a> <a href="rsspace.html#K13">l2_Space</a>  = 
 <a href="rsspace.html#K10">Zero_</a> <a href="rsspace.html#K11">the_set_of_l2RealSequences</a> ,<a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> 
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a>, <a class="ref" href="rsspace.html#D13">RSSPACE:def 13</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> 
<b>by </b><i><a class="ref" href="rsspace.html#D10">RSSPACE:def 10</a></i>
<br/>.= 
 <a href="rsspace.html#K6">Zeroseq</a> 
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a>, <a class="ref" href="rsspace.html#D7">RSSPACE:def 7</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E8:1"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a>  holds <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>2</sub></font></span>)</span> is <a href="series_1.html#V1">summable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a> ;<br/>

<a NAME="E1:1_8"/><i><font color="Green">E10</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="rsspace.html#D11">RSSPACE:def 11</a>, <a class="ref" href="rsspace.html#D13">RSSPACE:def 13</a></i>;<br/>
<a NAME="E2:1_8"/><i><font color="Green">E11</font></i>: 
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="rsspace.html#D11">RSSPACE:def 11</a>, <a class="ref" href="rsspace.html#D13">RSSPACE:def 13</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="seq_1.html#K22">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E3:1_8"/><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="rsspace2.html#E1:1_8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rsspace2.html#E2:1_8"><i><font color="Green">E11</font></i></a>, <a class="ref" href="series_1.html#T10">SERIES_1:10</a></i>;<br/>
<a NAME="E4:1_8"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<i><font color="Green">E14</font></i>: <span class="p1">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> = 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="complex1.html#K18">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">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:1_8_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 2 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<b>hence </b><a NAME="E4:1_8_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="rsspace2.html#E1:1_8_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xreal_1.html#T129">XREAL_1:129</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:1_8"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>12</sub></font> =  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<i><font color="Green">E14</font></i>: <span class="p1">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> = 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>
<br/>.= 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>10</sub></font></span>)</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">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T161">COMPLEX1:161</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T161">COMPLEX1:161</a></i>
;<br/>
<b>then </b><i><font color="Green">E15</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p5">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="rsspace2.html#E3:1_8_2"><i><font color="Green">E14</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 

;<br/>
<a NAME="E6:1_8_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E7:1_8_2"/><b>then </b>
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p5">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p5">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p5">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p0">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p5">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rsspace2.html#E5:1_8_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E8:1_8_2"/>
<span class="p1">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:1_8"/><b>then </b><i><font color="Green">E14</font></i>: 
2 <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="rsspace2.html#E3:1_8"><i><font color="Green">E12</font></i></a>, <a class="txt" href="rsspace2.html#E4:1_8"><i><font color="Green">E13</font></i></a>, <a class="ref" href="series_1.html#T24">SERIES_1:24</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = 1 <a href="real_1.html#K6">/</a> 2;<br/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p4">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></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#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T31">SEQ_1:31</a></i>
<br/>.= 
 <a href="seq_1.html#K22">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T35">SEQ_1:35</a></i>
;<br/>
<a NAME="E8:1_8"/><b>then </b>
 <a href="seq_1.html#K22">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="rsspace2.html#E6:1_8"><i><font color="Green">E14</font></i></a>, <a class="ref" href="series_1.html#T13">SERIES_1:13</a></i>;<br/>
<a NAME="E9:1_8"/><b>then </b>
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="series_1.html#V2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="series_1.html#D5">SERIES_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:1_8"/>
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="series_1.html#V1">summable</a>
 <b>by </b><i><a class="ref" href="series_1.html#T40">SERIES_1:40</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a> ;<br/>

<b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">c<sub>2</sub></font> = 
the <a href="bhsp_1.html#U1">scalar</a> of <a href="rsspace.html#K13">l2_Space</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>
<br/>.= 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace.html#D12">RSSPACE:def 12</a>, <a class="ref" href="rsspace.html#D13">RSSPACE:def 13</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:1"/>
( the <a href="struct_0.html#U1">carrier</a> of <a href="rsspace.html#K13">l2_Space</a>  <a href="hidden.html#R1">=</a>  <a href="rsspace.html#K11">the_set_of_l2RealSequences</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a>  iff ( <font color="Olive">b<sub>1</sub></font> is   <a href="seq_1.html#NM1">Real_Sequence</a> &amp; <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> is <a href="series_1.html#V1">summable</a> ) ) ) &amp;  <a href="rlvect_1.html#K1">0.</a> <a href="rsspace.html#K13">l2_Space</a>  <a href="hidden.html#R1">=</a>  <a href="rsspace.html#K6">Zeroseq</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>2</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>2</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a>  holds <br/>(  <a href="rlvect_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_1.html#K17">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp;  <a href="rsspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="seq_1.html#K17">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>2</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a>  holds <br/>( <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>2</sub></font></span>)</span> is <a href="series_1.html#V1">summable</a> &amp; ( for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace.html#K13">l2_Space</a>  holds <font color="Olive">b<sub>3</sub></font> <a href="bhsp_1.html#K1">.|.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>4</sub></font></span>)</span></span>)</span> ) ) ) )
 <b>by </b><i><a class="txt" href="rsspace2.html#E1:1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="rsspace2.html#E2:1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="rsspace2.html#E3:1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="rsspace2.html#E4:1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="rsspace2.html#E5:1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rsspace2.html#E6:1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="rsspace2.html#E7:1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="rsspace2.html#E8:1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="rsspace.html#D13">RSSPACE:def 13</a></i>;<br/>


</div>
