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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">s</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon" title="c2">s</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">^\</a> $1 is <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>;<br/>
<b>assume </b><a NAME="E1:41"/>
<font color="Maroon" title="c2">s</font> is <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>
 ;<br/>

<a NAME="E2:41"/><b>then </b><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="clopban3.html#T7" target="_self" title="CLOPBAN3:th.7">Th7</a></i>;<br/>
<a NAME="E3:41"/><i><font color="Green" title="E34">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:41_1"/>
<font color="Maroon" title="c2">s</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">^\</a> <font color="Maroon" title="c3">n</font> is <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>
 ;<br/>

<a NAME="E2:41_1"/><b>then </b>
 <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">^\</a> <font color="Maroon" title="c3">n</font></span>)</span> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a>
 
<b>by </b><i><a class="ref" href="clopban3.html#D2" target="_self" title="CLOPBAN3:def.2">Def2</a></i>;<br/>
<a NAME="E3:41_1"/><b>then </b><i><font color="Green" title="E35">A3</font></i>: 
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">^\</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">^\</a> 1 is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a>
 
<b>by </b><i><a class="ref" href="clopban3.html#T13" target="_self" title="CLOPBAN3:th.13">Th13</a></i>;<br/>
<a NAME="E4:41_1"/><i><font color="Green" title="E36">A4</font></i>: 
<font color="Maroon" title="c2">s</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">^\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">^\</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">^\</a> 1
 
<b>by </b><i><a class="ref" href="clopban3.html#T9" target="_self" title="CLOPBAN3:th.9">Th9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">s1</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">^\</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E6:41_1"/><i><font color="Green" title="E37">A5</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">s1</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">^\</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E7:41_1"/><i><font color="Green" title="E38">A6</font></i>: 
<font color="Maroon" title="c4">s1</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a>
 
<b>by </b><i><a class="ref" href="clopban3.html#T16" target="_self" title="CLOPBAN3:th.16">Th16</a></i>;<br/>
<a NAME="E8:41_1"/>
 <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">^\</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">^\</a> 1</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">^\</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">^\</a> 1</span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c4">s1</font>
 
<b>by </b><i><a class="txt" href="clopban3.html#E6:41_1"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="clopban3.html#T25" target="_self" title="CLOPBAN3:th.25">Th25</a></i>;<br/>
<a NAME="E9:41_1"/><b>then </b>
 <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">^\</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">^\</a> 1</span>)</span> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a>
 
<b>by </b><i><a class="txt" href="clopban3.html#E3:41_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="clopban3.html#E7:41_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="clvect_1.html#T116" title="CLVECT_1:th.116">CLVECT_1:116</a></i>;<br/>
<b>hence </b><a NAME="E10:41_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="clopban3.html#E4:41_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="clopban3.html#D2" target="_self" title="CLOPBAN3:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:41"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="clopban3.html#E2:41"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="clopban3.html#E3:41"><i><font color="Green" title="E34">A2</font></i></a>);</i><br/>


</div>
