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

<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:17"/><i><font color="Green" title="E13">A1</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">p</font>
 <b>and </b><br/><a NAME="E2:17"/><i><font color="Green" title="E14">A2</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">pp</font> = 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E4:17"/>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="holder_1.html#E1:17"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="real_2.html#T142" title="REAL_2:th.142">REAL_2:142</a></i>;<br/>
<a NAME="E5:17"/><b>then </b><i><font color="Green" title="E15">A3</font></i>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">pp</font>
 
<b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E6:17"/><b>then </b><i><font color="Green" title="E16">A4</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">q</font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E2:17"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="real_2.html#T150" title="REAL_2:th.150">REAL_2:150</a></i>;<br/>
<a NAME="E7:17"/><i><font color="Green" title="E17">A5</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="holder_1.html#E1:17"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">ap</font>, <font color="Maroon" title="c7">bq</font>, <font color="Maroon" title="c8">ab</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E8:17"/><i><font color="Green" title="E18">A6</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>  holds <br/> ( <font color="Maroon" title="c6">ap</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> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c7">bq</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> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font> &amp; <font color="Maroon" title="c8">ab</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>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E9:17"/><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c6">ap</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 <b>and </b><br/><a NAME="E10:17"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c7">bq</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<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>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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="c6">ap</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1</span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c9">x</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E12:17"/><i><font color="Green" title="E21">A9</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>  holds  <font color="Maroon" title="c9">x</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">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</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>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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="c7">bq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1</span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c10">y</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E14:17"/><i><font color="Green" title="E22">A10</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>  holds  <font color="Maroon" title="c10">y</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">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E15:17"/><i><font color="Green" title="E23">A11</font></i>: 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c6">ap</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="holder_1.html#E9:17"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a></i>;<br/>
<a NAME="E16:17"/><i><font color="Green" title="E24">A12</font></i>: 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c6">ap</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c6">ap</font></span>)</span>
 
<b>by </b><i><a class="ref" href="series_1.html#D3" title="SERIES_1:def.3">SERIES_1:def 3</a></i>;<br/>
<a NAME="E17:17"/>
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   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">ap</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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/>

<a NAME="E1:17_1"/><i><font color="Green" title="E25">A13</font></i>: 
<font color="Maroon" title="c6">ap</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E8:17"><i><font color="Green" title="E18">A6</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1"/>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1"><b>case </b></a><a NAME="E1:17_1_1_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_1_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">ap</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E1:17"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="holder_1.html#E1:17_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_2"><b>case </b></a><a NAME="E1:17_1_1_1_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_1_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">ap</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E1:17_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:17_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">ap</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E18:17"/><b>then </b>
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   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c6">ap</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E7"><i><font color="Green" title="E5">Lm2</font></i></a></i>;<br/>
<a NAME="E19:17"/><b>then </b><i><font color="Green" title="E25">A14</font></i>: 
( <font color="Maroon" title="c9">x</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c6">ap</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="holder_1.html#E7:17"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="holder_1.html#E12:17"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="holder_1.html#E15:17"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="holder_1.html#E16:17"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="holder_1.html#T10" target="_self" title="HOLDER_1:th.10">Th10</a></i>;<br/>
<a NAME="E20:17"/><i><font color="Green" title="E26">A15</font></i>: 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c7">bq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="holder_1.html#E10:17"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a></i>;<br/>
<a NAME="E21:17"/><i><font color="Green" title="E27">A16</font></i>: 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c7">bq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c7">bq</font></span>)</span>
 
<b>by </b><i><a class="ref" href="series_1.html#D3" title="SERIES_1:def.3">SERIES_1:def 3</a></i>;<br/>
<a NAME="E22:17"/>
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   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">bq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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/>

<a NAME="E1:17_2"/><i><font color="Green" title="E28">A17</font></i>: 
<font color="Maroon" title="c7">bq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c2">q</font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E8:17"><i><font color="Green" title="E18">A6</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_2_1_1"/>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_1"><b>case </b></a><a NAME="E1:17_2_1_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_2_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">bq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E6:17"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="holder_1.html#E1:17_2"><i><font color="Green" title="E28">A17</font></i></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_2_1_1_2"><b>case </b></a><a NAME="E1:17_2_1_1_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_2_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">bq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E1:17_2"><i><font color="Green" title="E28">A17</font></i></a>, <a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:17_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">bq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E23:17"/><b>then </b>
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   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c7">bq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E7"><i><font color="Green" title="E5">Lm2</font></i></a></i>;<br/>
<a NAME="E24:17"/><b>then </b>
( <font color="Maroon" title="c10">y</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c7">bq</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="holder_1.html#E2:17"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="holder_1.html#E5:17"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="holder_1.html#E14:17"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="holder_1.html#E20:17"><i><font color="Green" title="E26">A15</font></i></a>, <a class="txt" href="holder_1.html#E21:17"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="holder_1.html#T10" target="_self" title="HOLDER_1:th.10">Th10</a></i>;<br/>
<a NAME="E25:17"/><b>then </b><i><font color="Green" title="E28">A18</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c10">y</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c6">ap</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c7">bq</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="holder_1.html#E19:17"><i><font color="Green" title="E25">A14</font></i></a>, <a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a></i>;<br/>
<a NAME="E26:17"/><i><font color="Green" title="E29">A19</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>  holds  <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c8">ab</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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/>

<a NAME="E1:17_3"/><i><font color="Green" title="E30">A20</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c8">ab</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c6">ap</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c7">bq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="holder_1.html#E1:17"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="holder_1.html#E2:17"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="holder_1.html#E8:17"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="holder_1.html#T6" target="_self" title="HOLDER_1:th.6">Th6</a></i>;<br/>
<a NAME="E2:17_3"/><i><font color="Green" title="E31">A21</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c6">ap</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="holder_1.html#E12:17"><i><font color="Green" title="E21">A9</font></i></a></i>;<br/>
<a NAME="E3:17_3"/>
<font color="Maroon" title="c10">y</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">bq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="holder_1.html#E14:17"><i><font color="Green" title="E22">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:17_3"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c8">ab</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E1:17_3"><i><font color="Green" title="E30">A20</font></i></a>, <a class="txt" href="holder_1.html#E2:17_3"><i><font color="Green" title="E31">A21</font></i></a>, <a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:17"/>
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   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">ab</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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/>

<a NAME="E1:17_4"/>
<font color="Maroon" title="c8">ab</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="holder_1.html#E8:17"><i><font color="Green" title="E18">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:17_4"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">ab</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:17"/><b>then </b>
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c8">ab</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a>
 
<b>by </b><i><a class="ref" href="series_1.html#T19" title="SERIES_1:th.19">SERIES_1:19</a></i>;<br/>
<a NAME="E29:17"/><b>then </b>
(  <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c8">ab</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c8">ab</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c10">y</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="holder_1.html#E25:17"><i><font color="Green" title="E28">A18</font></i></a>, <a class="txt" href="holder_1.html#E26:17"><i><font color="Green" title="E29">A19</font></i></a>, <a class="ref" href="holder_1.html#T8" target="_self" title="HOLDER_1:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E30:17"/>
( <font color="Maroon" title="c8">ab</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp;  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c8">ab</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c6">ap</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c7">bq</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="holder_1.html#E25:17"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a>, <a class="ref" href="series_1.html#D3" title="SERIES_1:def.3">SERIES_1:def 3</a></i>;<br/>


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