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

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a> ;<br/>
<a NAME="E1:16"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  iff ( <font color="Olive">b<sub>1</sub></font> is   <a href="comseq_1.html#NM1">Complex_Sequence</a> &amp;  <a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font> is <a href="comseq_2.html#V1">bounded</a> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:16_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="csspace.html#K1">the_set_of_ComplexSequences</a>  iff <font color="Maroon">c<sub>2</sub></font> is   <a href="comseq_1.html#NM1">Complex_Sequence</a> )
 
<b>by </b><i><a class="ref" href="csspace.html#D1">CSSPACE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:16_1"/>
( <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  iff ( <font color="Maroon">c<sub>2</sub></font> is   <a href="comseq_1.html#NM1">Complex_Sequence</a> &amp;  <a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font> is <a href="comseq_2.html#V1">bounded</a> ) )
 <b>by </b><i><a class="ref" href="csspace4.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:16"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a> ;<br/>

<a NAME="E1:16_2"/>
<font color="Maroon">c<sub>2</sub></font> is   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a> 
 
<b>by </b><i><a class="txt" href="csspace4.html#E6"><i><font color="Green">Lemma6</font></i></a>, <a class="ref" href="clvect_1.html#T30">CLVECT_1:30</a></i>;<br/>
<b>hence </b><a NAME="E2:16_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="csspace.html#D2">CSSPACE:def 2</a>, <a class="ref" href="csspace.html#D7">CSSPACE:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:16"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a> ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a>  <b>by </b><i><a class="txt" href="csspace4.html#E6"><i><font color="Green">Lemma6</font></i></a>, <a class="ref" href="clvect_1.html#T30">CLVECT_1:30</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="csspace4.html#K1" target="_self">the_set_of_BoundedComplexSequences</a> ;<br/>
<a NAME="E2:16_3"/>
 <a href="relset_1.html#K4">dom</a> the <a href="rlvect_1.html#U1">add</a> of <a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:16_3"/><b>then </b><i><font color="Green">E18</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a>  <a href="realset1.html#K1">||</a> <a href="csspace4.html#K1" target="_self">the_set_of_BoundedComplexSequences</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="csspace4.html#K1" target="_self">the_set_of_BoundedComplexSequences</a> ,<a href="csspace4.html#K1" target="_self">the_set_of_BoundedComplexSequences</a> </span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a>  <a href="realset1.html#K1">||</a> <a href="csspace4.html#K1" target="_self">the_set_of_BoundedComplexSequences</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="csspace.html#D8">CSSPACE:def 8</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="csspace4.html#E3:16_3"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font>

;<br/>
<b>hence </b><a NAME="E5:16_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="csspace.html#T4">CSSPACE:4</a>, <a class="ref" href="csspace.html#D7">CSSPACE:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:16"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cfunct_1.html#NM1">Complex</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="comseq_1.html#K4">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a> ;<br/>



<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a>  <b>by </b><i><a class="txt" href="csspace4.html#E6"><i><font color="Green">Lemma6</font></i></a>, <a class="ref" href="clvect_1.html#T30">CLVECT_1:30</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="csspace4.html#K1" target="_self">the_set_of_BoundedComplexSequences</a> ;<br/>
<a NAME="E2:16_4"/><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<a href="csspace4.html#K1" target="_self">the_set_of_BoundedComplexSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E3:16_4"/>
 <a href="relset_1.html#K4">dom</a> the <a href="clvect_1.html#U1">Mult</a> of <a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:16_4"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="clvect_1.html#U1">Mult</a> of <a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<a href="csspace4.html#K1" target="_self">the_set_of_BoundedComplexSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<a href="csspace4.html#K1" target="_self">the_set_of_BoundedComplexSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="csspace4.html#E2:16_4"><i><font color="Green">E19</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<font color="Maroon">c<sub>2</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> = 
the <a href="clvect_1.html#U1">Mult</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D1">CLVECT_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="clvect_1.html#U1">Mult</a> of <a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<a href="csspace4.html#K1" target="_self">the_set_of_BoundedComplexSequences</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="csspace.html#D9">CSSPACE:def 9</a></i>
<br/>.= 
the <a href="clvect_1.html#U1">Mult</a> of <a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="csspace4.html#E4:16_4"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="clvect_1.html#D1">CLVECT_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E6:16_4"/>
<font color="Maroon">c<sub>2</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K4">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="csspace.html#T5">CSSPACE:5</a>, <a class="ref" href="csspace.html#D7">CSSPACE:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:16"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  holds <br/>(  <a href="rlvect_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="comseq_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp;  <a href="csspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="comseq_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a> ;<br/>

 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="clvect_1.html#T4">CLVECT_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="comseq_1.html#K4">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="csspace4.html#E4:16"><i><font color="Green">E18</font></i></a></i>
<br/>.= 
 <a href="comseq_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#T14">COMSEQ_1:14</a></i>
;<br/>
<b>hence </b><a NAME="E2:16_5"/>
(  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="comseq_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="csspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="comseq_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="csspace4.html#E2:16"><i><font color="Green">E16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:16"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a> ;<br/>

<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="csspace4.html#E3:16"><i><font color="Green">E17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="comseq_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="csspace4.html#E5:16"><i><font color="Green">E19</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D10">COMSEQ_1:def 10</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E7:16"/><i><font color="Green">E21</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  <a href="hidden.html#R1">=</a>  <a href="csspace.html#K6">CZeroseq</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_7"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="csspace4.html#K1" target="_self">the_set_of_BoundedComplexSequences</a> ;<br/>
<b>thus </b> <a href="rlvect_1.html#K1">0.</a> <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  = 
 <a href="csspace.html#K10">Zero_</a> <a href="csspace4.html#K1" target="_self">the_set_of_BoundedComplexSequences</a> ,<a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a> 

<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="csspace.html#K7">Linear_Space_of_ComplexSequences</a> 
<b>by </b><i><a class="ref" href="csspace.html#D10">CSSPACE:def 10</a></i>
<br/>.= 
 <a href="csspace.html#K6">CZeroseq</a> 
<b>by </b><i><a class="ref" href="csspace.html#D7">CSSPACE:def 7</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E8:16"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  holds <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a> ;<br/>

<b>thus </b><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_1.html#K5">.||</a></span> = 
the <a href="clvect_1.html#U2">norm</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="clvect_1.html#D9">CLVECT_1:def 9</a></i>
<br/>.= 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="csspace4.html#D2" target="_self">Def2</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:16"/>
( the <a href="struct_0.html#U1">carrier</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  <a href="hidden.html#R1">=</a>  <a href="csspace4.html#K1" target="_self">the_set_of_BoundedComplexSequences</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> is   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  iff ( <font color="Olive">b<sub>1</sub></font> is   <a href="comseq_1.html#NM1">Complex_Sequence</a> &amp;  <a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font> is <a href="comseq_2.html#V1">bounded</a> ) ) ) &amp;  <a href="rlvect_1.html#K1">0.</a> <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  <a href="hidden.html#R1">=</a>  <a href="csspace.html#K6">CZeroseq</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>2</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="cfunct_1.html#NM1">Complex</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="comseq_1.html#K4">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>2</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  holds <br/>(  <a href="rlvect_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="comseq_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp;  <a href="csspace.html#K2">seq_id</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="comseq_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="comseq_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>2</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  holds  <a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font> is <a href="comseq_2.html#V1">bounded</a> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace4.html#K3" target="_self">Complex_linfty_Space</a>  holds <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="csspace4.html#E1:16"><i><font color="Green">E15</font></i></a>, <a class="txt" href="csspace4.html#E2:16"><i><font color="Green">E16</font></i></a>, <a class="txt" href="csspace4.html#E3:16"><i><font color="Green">E17</font></i></a>, <a class="txt" href="csspace4.html#E4:16"><i><font color="Green">E18</font></i></a>, <a class="txt" href="csspace4.html#E5:16"><i><font color="Green">E19</font></i></a>, <a class="txt" href="csspace4.html#E6:16"><i><font color="Green">E20</font></i></a>, <a class="txt" href="csspace4.html#E7:16"><i><font color="Green">E21</font></i></a></i>;<br/>


</div>
