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

<span class="kw">set </span><font color="Maroon" title="c1">l1</font> =  <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a> ;<br/>
<a NAME="E1:17"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  iff ( <font color="Olive" title="b1">x</font> is   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> &amp;  <a href="csspace.html#K2" title="CSSPACE: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="17_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  iff ( <font color="Maroon" title="c2">x</font> is   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> &amp;  <a href="csspace.html#K2" title="CSSPACE: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>  ) )</span><br/>

<a NAME="E1:17_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a>  iff <font color="Maroon" title="c2">x</font> is   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#D1" title="CSSPACE:def.1">CSSPACE:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:17_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="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  iff ( <font color="Maroon" title="c2">x</font> is   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> &amp;  <a href="csspace.html#K2" title="CSSPACE: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>  ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="csspace4.html#D1" target="_self" title="CSSPACE4:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:17"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  holds  <font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0: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="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">v</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0: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="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span><br/>

<span class="kw">reconsider </span><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="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="csspace4.html#E6"><span class="lab"><font color="Green" title="E6">Lm5</font></span></a>, <a class="ref" href="clvect_1.html#T30" title="CLVECT_1:th.30">CLVECT_1:30</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">L1</font> =  <a href="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">W</font> =  <a href="csspace4.html#K1" title="CSSPACE4:func.1">the_set_of_BoundedComplexSequences</a> ;<br/>
<a NAME="E2:17_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</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="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:17_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="csspace4.html#K1" title="CSSPACE4:func.1">the_set_of_BoundedComplexSequences</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="csspace4.html#K1" title="CSSPACE4:func.1">the_set_of_BoundedComplexSequences</a> ,<a href="csspace4.html#K1" title="CSSPACE4:func.1">the_set_of_BoundedComplexSequences</a> </span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<font color="Maroon" title="c2">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">v</font> = 
<span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="csspace4.html#K1" title="CSSPACE4:func.1">the_set_of_BoundedComplexSequences</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#D8" title="CSSPACE:def.8">CSSPACE:def 8</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_2"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:17_2"/>
<font color="Maroon" title="c2">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0: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="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#T4" title="CSSPACE:th.4">CSSPACE:4</a>, <a class="ref" href="csspace.html#D7" title="CSSPACE:def.7">CSSPACE:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:17"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
for <font color="Olive" title="b1">c</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a><br/> for <font color="Olive" title="b2">u</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  holds  <font color="Olive" title="b1">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">c</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">u</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  holds  <font color="Maroon" title="c2">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">u</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">u</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c4">u1</font> = <font color="Maroon" title="c3">u</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="csspace4.html#E6"><span class="lab"><font color="Green" title="E6">Lm5</font></span></a>, <a class="ref" href="clvect_1.html#T30" title="CLVECT_1:th.30">CLVECT_1:30</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">L1</font> =  <a href="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">W</font> =  <a href="csspace4.html#K1" title="CSSPACE4:func.1">the_set_of_BoundedComplexSequences</a> ;<br/>
<a NAME="E2:17_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <a href="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</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#K2" title="NUMBERS:func.2">COMPLEX</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:17_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A5</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <a href="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</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#K2" title="NUMBERS:func.2">COMPLEX</a> ,<a href="csspace4.html#K1" title="CSSPACE4:func.1">the_set_of_BoundedComplexSequences</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#K2" title="NUMBERS:func.2">COMPLEX</a> ,<a href="csspace4.html#K1" title="CSSPACE4:func.1">the_set_of_BoundedComplexSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></span>;<br/>
<font color="Maroon" title="c2">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">u</font> = 
the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">u</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D1" title="CLVECT_1:def.1">CLVECT_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default">the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <a href="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</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#K2" title="NUMBERS:func.2">COMPLEX</a> ,<a href="csspace4.html#K1" title="CSSPACE4:func.1">the_set_of_BoundedComplexSequences</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">c</font>,<font color="Maroon" title="c3">u</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#D9" title="CSSPACE:def.9">CSSPACE:def 9</a></span>
<br/>.= 
the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <a href="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</a>  <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">c</font>,<font color="Maroon" title="c3">u</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_3"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<font color="Maroon" title="c2">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D1" title="CLVECT_1:def.1">CLVECT_1:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:17_3"/>
<font color="Maroon" title="c2">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#T5" title="CSSPACE:th.5">CSSPACE:5</a>, <a class="ref" href="csspace.html#D7" title="CSSPACE:def.7">CSSPACE:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:17"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
for <font color="Olive" title="b1">u</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  holds  <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">u</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span><br/>

<a NAME="E1:17_4"/>
<font color="Maroon" title="c2">u</font> is   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="csspace4.html#E6"><span class="lab"><font color="Green" title="E6">Lm5</font></span></a>, <a class="ref" href="clvect_1.html#T30" title="CLVECT_1:th.30">CLVECT_1:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:17_4"/>
<font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#D2" title="CSSPACE:def.2">CSSPACE:def 2</a>, <a class="ref" href="csspace.html#D7" title="CSSPACE:def.7">CSSPACE:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:17"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
for <font color="Olive" title="b1">u</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  holds <br/> (  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> &amp;  <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">u</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> &amp;  <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> )</span><br/>

 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">u</font> = 
<span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#T4" title="CLVECT_1:th.4">CLVECT_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E17">A4</font></span></a></span>
<br/>.= 
 <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_1.html#T14" title="COMSEQ_1:th.14">COMSEQ_1:14</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:17_5"/>
(  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> &amp;  <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E18">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:17"/><span class="lab"><font color="Green" title="E20">A8</font></span>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  holds  <font color="Olive" title="b1">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">v</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</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="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c2">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">v</font> = 
<span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E16">A2</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c3">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E19">A7</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:17"/><span class="lab"><font color="Green" title="E21">A9</font></span>: 
for <font color="Olive" title="b1">v</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  holds  <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">v</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span><br/>

<span class="kw">thus </span><span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> = 
the <a href="clvect_1.html#U2" title="CLVECT_1:sel.2">norm</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#D13" title="CLVECT_1:def.13">CLVECT_1:def 13</a></span>
<br/>.= 
 <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace4.html#D2" target="_self" title="CSSPACE4:def.2">Def2</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#D10" title="CSSPACE:def.10">CSSPACE:def 10</a></span>
<br/>.= 
 <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#D7" title="CSSPACE:def.7">CSSPACE:def 7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:17"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="csspace4.html#K1" title="CSSPACE4:func.1">the_set_of_BoundedComplexSequences</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="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  iff ( <font color="Olive" title="b1">x</font> is   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> &amp;  <a href="csspace.html#K2" title="CSSPACE: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="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a>  &amp; ( for <font color="Olive" title="b1">u</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  holds  <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="csspace.html#K2" title="CSSPACE: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="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  holds  <font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0: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="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">v</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">c</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a><br/> for <font color="Olive" title="b2">u</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  holds  <font color="Olive" title="b1">c</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE: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="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  holds <br/> (  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> &amp;  <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE: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="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  holds  <font color="Olive" title="b1">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="valued_1.html#K46" title="VALUED_1:func.46">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE: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="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  holds   <a href="csspace.html#K2" title="CSSPACE: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="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace4.html#K3" title="CSSPACE4:func.3">Complex_linfty_Space</a>  holds  <span class="p1"><a href="clvect_1.html#K6" title="CLVECT_1:func.6">||.</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="clvect_1.html#K6" title="CLVECT_1:func.6">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E6:17"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E7:17"><span class="lab"><font color="Green" title="E21">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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