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<b>let </b><font color="Maroon" title="c1">r</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>;<br/><b>let </b><font color="Maroon" title="c2">u</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace.html#K20" title="CSSPACE:func.20">Complex_l2_Space</a> ;<br/>



<b>reconsider </b><font color="Maroon" title="c3">u1</font> = <font color="Maroon" title="c2">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>  <b>by </b><i><a class="ref" href="clvect_1.html#T30" title="CLVECT_1:th.30">CLVECT_1:30</a>, <a class="ref" href="csspace.html#T15" title="CSSPACE:th.15">CSSPACE:15</a>, <a class="ref" href="csspace.html#D20" title="CSSPACE:def.20">CSSPACE:def 20</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">L1</font> =  <a href="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</a> ;<br/>
<b>set </b><font color="Maroon" title="c5">W</font> =  <a href="csspace.html#K11" title="CSSPACE:func.11">the_set_of_l2ComplexSequences</a> ;<br/>
<a NAME="E2:14"/><i><font color="Green" title="E15">A1</font></i>: 
<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="csspace.html#K11" title="CSSPACE:func.11">the_set_of_l2ComplexSequences</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#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>
 
<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="E3:14"/>
 <a href="relat_1.html#K1" title="RELAT_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>
 
<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="E4:14"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_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="csspace.html#K11" title="CSSPACE:func.11">the_set_of_l2ComplexSequences</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="csspace.html#K11" title="CSSPACE:func.11">the_set_of_l2ComplexSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="csspace2.html#E2:14"><i><font color="Green" title="E15">A1</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="c1">r</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font> = 
the <a href="clvect_1.html#U1" title="CLVECT_1:sel.1">Mult</a> of <a href="csspace.html#K20" title="CSSPACE:func.20">Complex_l2_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="c1">r</font>,<font color="Maroon" title="c2">u</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D1" title="CLVECT_1:def.1">CLVECT_1:def 1</a></i>
<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="csspace.html#K11" title="CSSPACE:func.11">the_set_of_l2ComplexSequences</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="c1">r</font>,<font color="Maroon" title="c2">u</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="ref" href="csspace.html#T14" title="CSSPACE:th.14">CSSPACE:14</a>, <a class="ref" href="csspace.html#D9" title="CSSPACE:def.9">CSSPACE:def 9</a>, <a class="ref" href="csspace.html#D20" title="CSSPACE:def.20">CSSPACE:def 20</a></i>
<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="c1">r</font>,<font color="Maroon" title="c2">u</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="csspace2.html#E4:14"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="csspace.html#D20" title="CSSPACE:def.20">CSSPACE:def 20</a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">u1</font>
<b>by </b><i><a class="ref" href="clvect_1.html#D1" title="CLVECT_1:def.1">CLVECT_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E6:14"/>
<font color="Maroon" title="c1">r</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="comseq_1.html#K4" title="COMSEQ_1:func.4">(#)</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>
 <b>by </b><i><a class="ref" href="csspace.html#T17" title="CSSPACE:th.17">CSSPACE:17</a></i>;<br/>


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