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<div class="add">

<b>let </b><font color="Maroon">seq</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:36"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 ;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">seq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">seq</font></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:36"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/>
<a NAME="E3:36"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:36_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>thus </b><span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">seq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><a href="complex1.html#K5">0c</a> </span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">seq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T130">COMPLEX1:130</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">seq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">seq</font></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">seq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">seq</font></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:36"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">seq</font></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> = 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">seq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>

<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
0
<b>by </b><i>, <a class="ref" href="complex1.html#T130">COMPLEX1:130</a></i>
;<br/><br/>


</div>
