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

<a NAME="E1:20"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="csspace.html#K18" title="CSSPACE:func.18">Complex_l2_Space</a> holds <br/>( <font color="Olive" title="b1">x</font> is   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> &amp; <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<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">x</font></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a></span>)</span> 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="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">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="c1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="csspace.html#K18" title="CSSPACE:func.18">Complex_l2_Space</a> implies ( <font color="Maroon" title="c1">x</font> is   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> &amp; <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a></span>)</span> is  <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>  ) )</span><br/>

<a NAME="E1:20_1"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<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="Maroon" title="c1">x</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a></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="csspace2.html#E19"><span class="lab"><font color="Green" title="E19">Lm19</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:20_1"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<font color="Maroon" title="c1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="csspace.html#K18" title="CSSPACE:func.18">Complex_l2_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="c1">x</font> is   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> &amp; <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a></span>)</span> is  <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>  )</span><br/>

<a NAME="E3:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#D11" title="CSSPACE:def.11">CSSPACE:def 11</a>, <a class="ref" href="csspace.html#D18" title="CSSPACE:def.18">CSSPACE:def 18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:20_1"/>
<font color="Maroon" title="c1">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a></span>)</span> is  <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a> </span><br/>

<a NAME="E5:20_1"/>
<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="Maroon" title="c1">x</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</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="Maroon" title="c1">x</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="csspace2.html#E17"><span class="lab"><font color="Green" title="E17">Lm17</font></span></a></span>;<br/>
<a NAME="E6:20_1"/><span class="kw">then </span>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<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="c1">x</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a></span>)</span></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="comseq_1.html#T46" title="COMSEQ_1:th.46">COMSEQ_1:46</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:20_1"/>
<span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a></span>)</span> 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="comseq_3.html#D9" title="COMSEQ_3:def.9">COMSEQ_3:def 9</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="E2:20"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> is   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> &amp; <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<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">x</font></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a></span>)</span> is  <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</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="csspace.html#K18" title="CSSPACE:func.18">Complex_l2_Space</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">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="c1">x</font> is   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> &amp; <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a></span>)</span> is  <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>  implies <font color="Maroon" title="c1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="csspace.html#K18" title="CSSPACE:func.18">Complex_l2_Space</a> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20_2"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c1">x</font> is   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>
 <span class="kw">and </span><br/><a NAME="E2:20_2"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
<span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a></span>)</span> is  <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</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="c1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="csspace.html#K18" title="CSSPACE:func.18">Complex_l2_Space</a></span><br/>

<a NAME="E3:20_2"/>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<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="c1">x</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a></span>)</span></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2"><span class="lab"><font color="Green" title="E23">A5</font></span></a></span>;<br/>
<a NAME="E4:20_2"/><span class="kw">then </span>
<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="Maroon" title="c1">x</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</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"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_1.html#T46" title="COMSEQ_1:th.46">COMSEQ_1:46</a></span>;<br/>
<a NAME="E5:20_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<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="Maroon" title="c1">x</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</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="Maroon" title="c1">x</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="csspace2.html#E19"><span class="lab"><font color="Green" title="E19">Lm19</font></span></a></span>;<br/>
<a NAME="E6:20_2"/>
<font color="Maroon" title="c1">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_2"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="csspace.html#D1" title="CSSPACE:def.1">CSSPACE:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:20_2"/>
<font color="Maroon" title="c1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="csspace.html#K18" title="CSSPACE:func.18">Complex_l2_Space</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_2"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="csspace.html#D11" title="CSSPACE:def.11">CSSPACE:def 11</a>, <a class="ref" href="csspace.html#D18" title="CSSPACE:def.18">CSSPACE:def 18</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>
<span class="kw">hence </span><a NAME="E3:20"/>
 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="csspace.html#K18" title="CSSPACE:func.18">Complex_l2_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; <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p1">(<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">x</font></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a></span>)</span> 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="txt" href="#E1:20"><span class="lab"><font color="Green" title="E21">A1</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>
