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

<span class="kw">set </span><font color="Maroon" title="c1">l1</font> =  <a href="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_Space</a> ;<br/>
<a NAME="E1:11"/><span class="lab"><font color="Green" title="E10">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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_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="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_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="firebrick">::  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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_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="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>  ) )</span><br/>

<a NAME="E1:11_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="E2:11_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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_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="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="csspace3.html#D1" target="_self" title="CSSPACE3:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:11"/><span class="lab"><font color="Green" title="E11">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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_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="series_1.html#K1" title="SERIES_1:func.1">+</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="11_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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_Space</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="series_1.html#K1" title="SERIES_1:func.1">+</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="csspace3.html#E3"><span class="lab"><font color="Green" title="E3">Lm1</font></span></a>, <a class="ref" href="clvect_1.html#T29" title="CLVECT_1:th.29">CLVECT_1:29</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="csspace3.html#K1" title="CSSPACE3:func.1">the_set_of_l1ComplexSequences</a> ;<br/>
<a NAME="E2:11_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="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="E3:11_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">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="csspace3.html#K1" title="CSSPACE3:func.1">the_set_of_l1ComplexSequences</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="csspace3.html#K1" title="CSSPACE3:func.1">the_set_of_l1ComplexSequences</a>,<a href="csspace3.html#K1" title="CSSPACE3:func.1">the_set_of_l1ComplexSequences</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#T62" title="RELAT_1:th.62">RELAT_1:62</a>, <a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</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="csspace3.html#K1" title="CSSPACE3:func.1">the_set_of_l1ComplexSequences</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="#E3:11_2"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:11_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="series_1.html#K1" title="SERIES_1:func.1">+</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#T2" title="CSSPACE:th.2">CSSPACE: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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:11"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
 for <font color="Olive" title="b1">p</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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_Space</a> holds  <font color="Olive" title="b1">p</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">p</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="11_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">p</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="firebrick">::  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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_Space</a> holds  <font color="Maroon" title="c2">p</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">p</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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_Space</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</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">p</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="csspace3.html#E3"><span class="lab"><font color="Green" title="E3">Lm1</font></span></a>, <a class="ref" href="clvect_1.html#T29" title="CLVECT_1:th.29">CLVECT_1:29</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="csspace3.html#K1" title="CSSPACE3:func.1">the_set_of_l1ComplexSequences</a> ;<br/>
<a NAME="E2:11_3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="E3:11_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A5</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="csspace3.html#K1" title="CSSPACE3:func.1">the_set_of_l1ComplexSequences</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="csspace3.html#K1" title="CSSPACE3:func.1">the_set_of_l1ComplexSequences</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#T62" title="RELAT_1:th.62">RELAT_1:62</a>, <a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></span>;<br/>
<font color="Maroon" title="c2">p</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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_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">p</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="csspace3.html#K1" title="CSSPACE3:func.1">the_set_of_l1ComplexSequences</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">p</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">p</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="#E3:11_3"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<font color="Maroon" title="c2">p</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="E5:11_3"/>
<font color="Maroon" title="c2">p</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">p</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#T3" title="CSSPACE:th.3">CSSPACE:3</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:11"/><span class="lab"><font color="Green" title="E13">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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_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="11_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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_Space</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E1:11_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="csspace3.html#E3"><span class="lab"><font color="Green" title="E3">Lm1</font></span></a>, <a class="ref" href="clvect_1.html#T29" title="CLVECT_1:th.29">CLVECT_1:29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:11_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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:11"/><span class="lab"><font color="Green" title="E14">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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_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="11_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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_Space</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="csspace3.html#T5" target="_self" title="CSSPACE3:th.5">Th7</a>, <a class="ref" href="clvect_1.html#T3" title="CLVECT_1:th.3">CLVECT_1:3</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="#E3:11"><span class="lab"><font color="Green" title="E12">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#T11" title="COMSEQ_1:th.11">COMSEQ_1:11</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:11_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="#E4:11"><span class="lab"><font color="Green" title="E13">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:11"/><span class="lab"><font color="Green" title="E15">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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_Space</a> holds  <font color="Olive" title="b1">u</font> <a href="algstr_0.html#K5" title="ALGSTR_0: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="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="11_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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_Space</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">u</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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="series_1.html#K1" title="SERIES_1:func.1">+</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="#E2:11"><span class="lab"><font color="Green" title="E11">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="#E5:11"><span class="lab"><font color="Green" title="E14">A7</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:11"/><span class="lab"><font color="Green" title="E16">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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_Space</a> holds  <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<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 class="kw">by</span> <span class="lab"><a class="ref" href="csspace3.html#D2" target="_self" title="CSSPACE3:def.2">Def2</a></span>;<br/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_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="E9:11"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_Space</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="csspace3.html#K1" title="CSSPACE3:func.1">the_set_of_l1ComplexSequences</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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_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="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>  ) ) ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_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="series_1.html#K1" title="SERIES_1:func.1">+</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">p</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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_Space</a> holds  <font color="Olive" title="b1">p</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">p</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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_Space</a> holds  <font color="Olive" title="b1">u</font> <a href="algstr_0.html#K5" title="ALGSTR_0: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="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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_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="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</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="csspace3.html#K3" title="CSSPACE3:func.3">Complex_l1_Space</a> holds  <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E5:11"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="txt" href="#E6:11"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="txt" href="#E7:11"><span class="lab"><font color="Green" title="E16">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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