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

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="rsspace3.html#K3" target="_self">l1_Space</a> ;<br/>
<a NAME="E1:13"/><i><font color="Green">E12</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="rsspace3.html#K3" target="_self">l1_Space</a>  iff ( <font color="Olive">b<sub>1</sub></font> is   <a href="seq_1.html#NM1">Real_Sequence</a> &amp;  <a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font> is <a href="series_1.html#V2">absolutely_summable</a> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:13_1"/>
( <font color="Maroon">c<sub>2</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>2</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="E2:13_1"/>
( <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <a href="rsspace3.html#K3" target="_self">l1_Space</a>  iff ( <font color="Maroon">c<sub>2</sub></font> is   <a href="seq_1.html#NM1">Real_Sequence</a> &amp;  <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font> is <a href="series_1.html#V2">absolutely_summable</a> ) )
 <b>by </b><i><a class="ref" href="rsspace3.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:13"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace3.html#K3" target="_self">l1_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="13_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace3.html#K3" target="_self">l1_Space</a> ;<br/>

<a NAME="E1:13_2"/>
<font color="Maroon">c<sub>2</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="txt" href="rsspace3.html#E3"><i><font color="Green">Lemma3</font></i></a>, <a class="ref" href="rlsub_1.html#T18">RLSUB_1:18</a></i>;<br/>
<b>hence </b><a NAME="E2:13_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rsspace.html#D2">RSSPACE:def 2</a>, <a class="ref" href="rsspace.html#D7">RSSPACE:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:13"/><i><font color="Green">E14</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="rsspace3.html#K3" target="_self">l1_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</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="13_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace3.html#K3" target="_self">l1_Space</a> ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</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="txt" href="rsspace3.html#E3"><i><font color="Green">Lemma3</font></i></a>, <a class="ref" href="rlsub_1.html#T18">RLSUB_1:18</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="rsspace.html#K7">Linear_Space_of_RealSequences</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="rsspace3.html#K1" target="_self">the_set_of_l1RealSequences</a> ;<br/>
<a NAME="E2:13_3"/><i><font color="Green">E15</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="rsspace3.html#K1" target="_self">the_set_of_l1RealSequences</a> ,<a href="rsspace3.html#K1" target="_self">the_set_of_l1RealSequences</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="E3:13_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="E4:13_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="rsspace3.html#K1" target="_self">the_set_of_l1RealSequences</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="rsspace3.html#K1" target="_self">the_set_of_l1RealSequences</a> ,<a href="rsspace3.html#K1" target="_self">the_set_of_l1RealSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="rsspace3.html#E2:13_3"><i><font color="Green">E15</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E5:13_3"/><b>then </b><i><font color="Green">E16</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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="rsspace3.html#K1" target="_self">the_set_of_l1RealSequences</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <a href="rsspace3.html#K3" target="_self">l1_Space</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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="rsspace3.html#K1" target="_self">the_set_of_l1RealSequences</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>2</sub></font>,<font color="Maroon">c<sub>3</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></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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="rsspace3.html#E5:13_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</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="E7:13_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</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>2</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rsspace.html#T4">RSSPACE:4</a>, <a class="ref" href="rsspace.html#D7">RSSPACE:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:13"/><i><font color="Green">E15</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="rsspace3.html#K3" target="_self">l1_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="13_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace3.html#K3" target="_self">l1_Space</a> ;<br/>



<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</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="txt" href="rsspace3.html#E3"><i><font color="Green">Lemma3</font></i></a>, <a class="ref" href="rlsub_1.html#T18">RLSUB_1:18</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="rsspace3.html#K1" target="_self">the_set_of_l1RealSequences</a> ;<br/>
<a NAME="E2:13_4"/><i><font color="Green">E16</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="rsspace3.html#K1" target="_self">the_set_of_l1RealSequences</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="E3:13_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="E4:13_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="rsspace3.html#K1" target="_self">the_set_of_l1RealSequences</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="rsspace3.html#K1" target="_self">the_set_of_l1RealSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="rsspace3.html#E2:13_4"><i><font color="Green">E16</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E5:13_4"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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="rsspace3.html#K1" target="_self">the_set_of_l1RealSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font> = 
the <a href="rlvect_1.html#U2">Mult</a> of <a href="rsspace3.html#K3" target="_self">l1_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>2</sub></font>,<font color="Maroon">c<sub>3</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="rsspace3.html#K1" target="_self">the_set_of_l1RealSequences</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>2</sub></font>,<font color="Maroon">c<sub>3</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></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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="rsspace3.html#E5:13_4"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</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="E7:13_4"/>
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rsspace.html#T5">RSSPACE:5</a>, <a class="ref" href="rsspace.html#D7">RSSPACE:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:13"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace3.html#K3" target="_self">l1_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="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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace3.html#K3" target="_self">l1_Space</a> ;<br/>

 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>2</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>2</sub></font>
<b>by </b><i><a class="ref" href="rsspace3.html#T7" target="_self">Th7</a>, <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>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rsspace3.html#E4:13"><i><font color="Green">E15</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>2</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="E2:13_5"/>
(  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>2</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>2</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>2</sub></font></span>)</span> <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>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="rsspace3.html#E2:13"><i><font color="Green">E13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:13"/><i><font color="Green">E17</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="rsspace3.html#K3" target="_self">l1_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="13_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace3.html#K3" target="_self">l1_Space</a> ;<br/>

<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>3</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>2</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>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rsspace3.html#E3:13"><i><font color="Green">E14</font></i></a></i>
<br/>.= 
<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#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>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rsspace3.html#E5:13"><i><font color="Green">E16</font></i></a></i>
<br/>.= 
<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#K10">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</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="E7:13"/><i><font color="Green">E18</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <a href="rsspace3.html#K3" target="_self">l1_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="13_7"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="rsspace3.html#K1" target="_self">the_set_of_l1RealSequences</a> ;<br/>
<b>thus </b> <a href="rlvect_1.html#K1">0.</a> <a href="rsspace3.html#K3" target="_self">l1_Space</a>  = 
 <a href="rsspace.html#K10">Zero_</a> <a href="rsspace3.html#K1" target="_self">the_set_of_l1RealSequences</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></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="E8:13"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace3.html#K3" target="_self">l1_Space</a>  holds <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace3.html#K3" target="_self">l1_Space</a> ;<br/>

<b>thus </b><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="normsp_1.html#K1">.||</a></span> = 
the <a href="normsp_1.html#U1">norm</a> of <a href="rsspace3.html#K3" target="_self">l1_Space</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="normsp_1.html#D1">NORMSP_1:def 1</a></i>
<br/>.= 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rsspace3.html#D2" target="_self">Def2</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:13"/>
( the <a href="struct_0.html#U1">carrier</a> of <a href="rsspace3.html#K3" target="_self">l1_Space</a>  <a href="hidden.html#R1">=</a>  <a href="rsspace3.html#K1" target="_self">the_set_of_l1RealSequences</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="rsspace3.html#K3" target="_self">l1_Space</a>  iff ( <font color="Olive">b<sub>1</sub></font> is   <a href="seq_1.html#NM1">Real_Sequence</a> &amp;  <a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font> is <a href="series_1.html#V2">absolutely_summable</a> ) ) ) &amp;  <a href="rlvect_1.html#K1">0.</a> <a href="rsspace3.html#K3" target="_self">l1_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="rsspace3.html#K3" target="_self">l1_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="rsspace3.html#K3" target="_self">l1_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</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="rsspace3.html#K3" target="_self">l1_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="rsspace3.html#K3" target="_self">l1_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="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; ( 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="rsspace3.html#K3" target="_self">l1_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> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace3.html#K3" target="_self">l1_Space</a>  holds  <a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font> is <a href="series_1.html#V2">absolutely_summable</a> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace3.html#K3" target="_self">l1_Space</a>  holds <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="rsspace3.html#E1:13"><i><font color="Green">E12</font></i></a>, <a class="txt" href="rsspace3.html#E2:13"><i><font color="Green">E13</font></i></a>, <a class="txt" href="rsspace3.html#E3:13"><i><font color="Green">E14</font></i></a>, <a class="txt" href="rsspace3.html#E4:13"><i><font color="Green">E15</font></i></a>, <a class="txt" href="rsspace3.html#E5:13"><i><font color="Green">E16</font></i></a>, <a class="txt" href="rsspace3.html#E6:13"><i><font color="Green">E17</font></i></a>, <a class="txt" href="rsspace3.html#E7:13"><i><font color="Green">E18</font></i></a></i>;<br/>


</div>
