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<div class="add">

<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon" title="c2">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><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="E1:35"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; ( 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="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</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> <a href="series_1.html#K5" title="SERIES_1:func.5">Sum</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">N</font>,$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> $1</span>)</span>;<br/>
<a NAME="E2:35"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

 <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Maroon" title="c1">f</font>,0 = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
<b>by </b><i><a class="ref" href="series_1.html#D6" title="SERIES_1:def.6">SERIES_1:def 6</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="asympt_1.html#E1:35"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>
;<br/>
<b>then </b> <a href="series_1.html#K5" title="SERIES_1:func.5">Sum</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">N</font>,0 = 
<span class="p1">(<span class="default"><a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 0
<b>by </b><i><a class="ref" href="series_1.html#D7" title="SERIES_1:def.7">SERIES_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E1:35"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="asympt_1.html#E28"><i><font color="Green" title="E24">Lm19</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E3:35_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:35"/><i><font color="Green" title="E31">A3</font></i>: 
for <font color="Olive" title="b1">M</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">M</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">M</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">M</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="E1:35_2"/><i><font color="Green" title="E32">A4</font></i>: 
 <a href="series_1.html#K5" title="SERIES_1:func.5">Sum</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c4">M</font></span>)</span>
 ;<br/>

 <a href="series_1.html#K5" title="SERIES_1:func.5">Sum</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">N</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D7" title="SERIES_1:def.7">SERIES_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D6" title="SERIES_1:def.6">SERIES_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">N</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">M</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D6" title="SERIES_1:def.6">SERIES_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E1:35_2"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="series_1.html#D7" title="SERIES_1:def.7">SERIES_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">b</font></span>)</span>
<b>by </b><i><a class="txt" href="asympt_1.html#E1:35"><i><font color="Green" title="E29">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E3:35_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">M</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:35"/>
for <font color="Olive" title="b1">M</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">M</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="asympt_1.html#E2:35"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="asympt_1.html#E3:35"><i><font color="Green" title="E31">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:35"/>
for <font color="Olive" title="b1">M</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   <a href="series_1.html#K5" title="SERIES_1:func.5">Sum</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">N</font>,<font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b1">M</font></span>)</span>
 ;<br/>


</div>
