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

<b>set </b><font color="Maroon" title="c1">l1</font> =  <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a> ;<br/>
<a NAME="E1:12"/><i><font color="Green" title="E11">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  iff ( <font color="Olive" title="b1">x</font> is   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> &amp;  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">x</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E1:12_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rsspace.html#K1" title="RSSPACE:func.1">the_set_of_RealSequences</a>  iff <font color="Maroon" title="c2">x</font> is   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> )
 
<b>by </b><i><a class="ref" href="rsspace.html#D1" title="RSSPACE:def.1">RSSPACE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:12_1"/>
( <font color="Maroon" title="c2">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  iff ( <font color="Maroon" title="c2">x</font> is   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> &amp;  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">x</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> ) )
 <b>by </b><i><a class="ref" href="rsspace4.html#D1" target="_self" title="RSSPACE4:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:12"/><i><font color="Green" title="E12">A2</font></i>: 
for <font color="Olive" title="b1">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  holds  <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a> ;<br/>

<a NAME="E1:12_2"/>
<font color="Maroon" title="c2">u</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> 
 
<b>by </b><i><a class="txt" href="rsspace4.html#E4"><i><font color="Green" title="E4">Lm3</font></i></a>, <a class="ref" href="rlsub_1.html#T18" title="RLSUB_1:th.18">RLSUB_1:18</a></i>;<br/>
<b>hence </b><a NAME="E2:12_2"/>
<font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font>
 <b>by </b><i><a class="ref" href="rsspace.html#D2" title="RSSPACE:def.2">RSSPACE:def 2</a>, <a class="ref" href="rsspace.html#D7" title="RSSPACE:def.7">RSSPACE:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:12"/><i><font color="Green" title="E13">A3</font></i>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  holds  <font color="Olive" title="b1">u</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">v</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a> ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">u1</font> = <font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c5">v1</font> = <font color="Maroon" title="c3">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  <b>by </b><i><a class="txt" href="rsspace4.html#E4"><i><font color="Green" title="E4">Lm3</font></i></a>, <a class="ref" href="rlsub_1.html#T18" title="RLSUB_1:th.18">RLSUB_1:18</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">L1</font> =  <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> ;<br/>
<b>set </b><font color="Maroon" title="c7">W</font> =  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> ;<br/>
<a NAME="E2:12_3"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:12_3"/><b>then </b><i><font color="Green" title="E14">A4</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> ,<a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> </span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<font color="Maroon" title="c2">u</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c3">v</font> = 
<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="ref" href="rsspace.html#D8" title="RSSPACE:def.8">RSSPACE:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">v1</font>
<b>by </b><i><a class="txt" href="rsspace4.html#E3:12_3"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
;<br/>
<b>hence </b><a NAME="E5:12_3"/>
<font color="Maroon" title="c2">u</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span>
 <b>by </b><i><a class="ref" href="rsspace.html#T4" title="RSSPACE:th.4">RSSPACE:4</a>, <a class="ref" href="rsspace.html#D7" title="RSSPACE:def.7">RSSPACE:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:12"/><i><font color="Green" title="E14">A5</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  holds  <font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c3">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a> ;<br/>



<b>reconsider </b><font color="Maroon" title="c4">u1</font> = <font color="Maroon" title="c3">u</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  <b>by </b><i><a class="txt" href="rsspace4.html#E4"><i><font color="Green" title="E4">Lm3</font></i></a>, <a class="ref" href="rlsub_1.html#T18" title="RLSUB_1:th.18">RLSUB_1:18</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">L1</font> =  <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> ;<br/>
<b>set </b><font color="Maroon" title="c6">W</font> =  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> ;<br/>
<a NAME="E2:12_4"/><i><font color="Green" title="E15">A6</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E3:12_4"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:12_4"/><b>then </b><i><font color="Green" title="E16">A7</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="rsspace4.html#E2:12_4"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<font color="Maroon" title="c2">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">u</font> = 
<span class="p1">(<span class="default">the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">u</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="ref" href="rsspace.html#D9" title="RSSPACE:def.9">RSSPACE:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">u1</font>
<b>by </b><i><a class="txt" href="rsspace4.html#E4:12_4"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
;<br/>
<b>hence </b><a NAME="E6:12_4"/>
<font color="Maroon" title="c2">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="rsspace.html#T5" title="RSSPACE:th.5">RSSPACE:5</a>, <a class="ref" href="rsspace.html#D7" title="RSSPACE:def.7">RSSPACE:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:12"/><i><font color="Green" title="E15">A8</font></i>: 
for <font color="Olive" title="b1">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  holds <br/> (  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_1.html#K17" title="SEQ_1:func.17">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> &amp;  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="seq_1.html#K17" title="SEQ_1:func.17">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a> ;<br/>

 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">u</font> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c2">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T29" title="RLVECT_1:th.29">RLVECT_1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span>
<b>by </b><i><a class="txt" href="rsspace4.html#E4:12"><i><font color="Green" title="E14">A5</font></i></a></i>
<br/>.= 
 <a href="seq_1.html#K17" title="SEQ_1:func.17">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T25" title="SEQ_1:th.25">SEQ_1:25</a></i>
;<br/>
<b>hence </b><a NAME="E2:12_5"/>
(  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_1.html#K17" title="SEQ_1:func.17">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> &amp;  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="seq_1.html#K17" title="SEQ_1:func.17">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rsspace4.html#E2:12"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:12"/><i><font color="Green" title="E16">A9</font></i>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  holds  <font color="Olive" title="b1">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">v</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a> ;<br/>

<b>thus </b><font color="Maroon" title="c2">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c3">v</font> = 
<span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rsspace4.html#E3:12"><i><font color="Green" title="E13">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17" title="SEQ_1:func.17">-</a> <span class="p2">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rsspace4.html#E5:12"><i><font color="Green" title="E15">A8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T15" title="SEQ_1:th.15">SEQ_1:15</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E7:12"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rsspace.html#K6" title="RSSPACE:func.6">Zeroseq</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_7"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c2">W</font> =  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a> ;<br/>
<b>thus </b> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rsspace.html#K7" title="RSSPACE:func.7">Linear_Space_of_RealSequences</a> 
<b>by </b><i><a class="ref" href="rsspace.html#D10" title="RSSPACE:def.10">RSSPACE:def 10</a></i>
<br/>.= 
 <a href="rsspace.html#K6" title="RSSPACE:func.6">Zeroseq</a> 
<b>by </b><i><a class="ref" href="rsspace.html#D7" title="RSSPACE:def.7">RSSPACE:def 7</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:12"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rsspace4.html#K1" title="RSSPACE4:func.1">the_set_of_BoundedRealSequences</a>  &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  iff ( <font color="Olive" title="b1">x</font> is   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> &amp;  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">x</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> ) ) ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rsspace.html#K6" title="RSSPACE:func.6">Zeroseq</a>  &amp; ( for <font color="Olive" title="b1">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  holds  <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font> ) &amp; ( for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  holds  <font color="Olive" title="b1">u</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">v</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  holds  <font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">u</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">u</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  holds <br/> (  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_1.html#K17" title="SEQ_1:func.17">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> &amp;  <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="seq_1.html#K17" title="SEQ_1:func.17">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> ) ) &amp; ( for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  holds  <font color="Olive" title="b1">u</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p1">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">v</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  holds   <a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">v</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> ) &amp; ( for <font color="Olive" title="b1">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="rsspace4.html#K3" title="RSSPACE4:func.3">linfty_Space</a>  holds  <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22" title="SEQ_1:func.22">abs</a> <span class="p3">(<span class="default"><a href="rsspace.html#K2" title="RSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">v</font></span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="rsspace4.html#E1:12"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="rsspace4.html#E2:12"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="rsspace4.html#E3:12"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="rsspace4.html#E4:12"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="rsspace4.html#E5:12"><i><font color="Green" title="E15">A8</font></i></a>, <a class="txt" href="rsspace4.html#E6:12"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="rsspace4.html#D2" target="_self" title="RSSPACE4:def.2">Def2</a></i>;<br/>


</div>
