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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:11"/><i><font color="Green">E9</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:11"/><i><font color="Green">E10</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = 1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>4</sub></font> = 1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E4:11"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E2:11"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E5:11"/>
1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="holder_1.html#E1:11"><i><font color="Green">E9</font></i></a>, <a class="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/>
<a NAME="E6:11"/><b>then </b>
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E7:11"/><b>then </b><i><font color="Green">E12</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E4:11"><i><font color="Green">E11</font></i></a>, <a class="ref" href="real_2.html#T150">REAL_2:150</a></i>;<br/>
<a NAME="E8:11"/><i><font color="Green">E13</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E1:11"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E9:11"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="holder_1.html#E8:11"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E10:11"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="holder_1.html#E7:11"><i><font color="Green">E12</font></i></a></i>;<br/>
<a NAME="E11:11"/><i><font color="Green">E16</font></i>: 
1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="holder_1.html#E8:11"><i><font color="Green">E13</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E12:11"/><i><font color="Green">E17</font></i>: 
1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="holder_1.html#E7:11"><i><font color="Green">E12</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E13:11"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E14:11"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:11_1"/><i><font color="Green">E20</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E2:11_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E13:11"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E3:11_1"/><i><font color="Green">E22</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 or  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="holder_1.html#E1:11_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1_1_1"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 or  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="holder_1.html#E3:11_1"><i><font color="Green">E22</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_1"><b>case </b></a><a NAME="E1:11_1_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="holder_1.html#E8:11"><i><font color="Green">E13</font></i></a>, <a class="txt" href="holder_1.html#E2:11_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_2"><b>case </b></a><a NAME="E1:11_1_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="holder_1.html#E2:11_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="power.html#T39">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="E5:11_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:11"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:11_2"/><i><font color="Green">E21</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E2:11_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E13:11"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E3:11_2"/><i><font color="Green">E23</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 or  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="holder_1.html#E1:11_2"><i><font color="Green">E21</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_2_1_1"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 or  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="holder_1.html#E3:11_2"><i><font color="Green">E23</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_1_1_1"><b>case </b></a><a NAME="E1:11_2_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_2_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="holder_1.html#E7:11"><i><font color="Green">E12</font></i></a>, <a class="txt" href="holder_1.html#E2:11_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_1_1_2"><b>case </b></a><a NAME="E1:11_2_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_2_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="holder_1.html#E2:11_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="power.html#T39">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="E5:11_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E16:11"/><i><font color="Green">E21</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E14:11"><i><font color="Green">E19</font></i></a>, <a class="txt" href="holder_1.html#E7"><i><font color="Green">Lemma5</font></i></a></i>;<br/>
<a NAME="E17:11"/><i><font color="Green">E22</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E15:11"><i><font color="Green">E20</font></i></a>, <a class="txt" href="holder_1.html#E7"><i><font color="Green">Lemma5</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_3_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_1"><b>case </b></a><a NAME="E1:11_3_1_1"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:11_3_1_1"/><i><font color="Green">E24</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:11_3_1_1_1"/><i><font color="Green">E25</font></i>: 
( <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="holder_1.html#E16:11"><i><font color="Green">E21</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_3_1_1_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> or 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> )
 </span><b>by </b><i><a class="txt" href="holder_1.html#E1:11_3_1_1_1"><i><font color="Green">E25</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:11_3_1_1_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_3_1_1_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:11_3_1_1_1_1_1_2"/>
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><a NAME="E2:11_3_1_1_1_1_1_2"/><b>then </b>
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E11:11"><i><font color="Green">E16</font></i></a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><b>hence </b><a NAME="E3:11_3_1_1_1_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:11_3_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:11_3_1_1"/><i><font color="Green">E25</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:11_3_1_1_2"/><i><font color="Green">E26</font></i>: 
( <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="holder_1.html#E17:11"><i><font color="Green">E22</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_3_1_1_2_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> or 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> )
 </span><b>by </b><i><a class="txt" href="holder_1.html#E1:11_3_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:11_3_1_1_2_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_3_1_1_2_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:11_3_1_1_2_1_1_2"/><i><font color="Green">E27</font></i>: 
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><a NAME="E2:11_3_1_1_2_1_1_2"/>
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E12:11"><i><font color="Green">E17</font></i></a>, <a class="txt" href="holder_1.html#E1:11_3_1_1_2_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><b>hence </b><a NAME="E3:11_3_1_1_2_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:11_3_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E4:11_3_1_1"/><i><font color="Green">E26</font></i>: 
0 <a href="xxreal_0.html#R1">&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">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</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">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E2:11_3_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="holder_1.html#E3:11_3_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/><a NAME="E5:11_3_1_1"/><i><font color="Green">E27</font></i>: 
( <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0 or <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="holder_1.html#E1:11_3_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_3_1_1_3_1"/>
( <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0 or <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i><a class="txt" href="holder_1.html#E5:11_3_1_1"><i><font color="Green">E27</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_1_3_1_1"><b>case </b></a><a NAME="E1:11_3_1_1_3_1_1"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:11_3_1_1_3_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="holder_1.html#E14:11"><i><font color="Green">E19</font></i></a>, <a class="txt" href="holder_1.html#E1:11_3_1_1_3_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="holder_1.html#E8"><i><font color="Green">Lemma6</font></i></a></i>;<br/><a NAME="E3:11_3_1_1_3_1_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:11_3_1_1_3_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:11_3_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="holder_1.html#E2:11_3_1_1_3_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="holder_1.html#E1:11_3_1_1_3_1_1_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E3:11_3_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="holder_1.html#E13:11"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E4:11_3_1_1_3_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_3_1_1_3_1_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:11_3_1_1_3_1_1_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<b>hence </b><a NAME="E3:11_3_1_1_3_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="holder_1.html#E3:11_3_1_1_3_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:11_3_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:11_3_1_1_3_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1_3_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:11_3_1_1_3_1_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>thus </b><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E13:11"><i><font color="Green">E18</font></i></a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E3:11_3_1_1_3_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="holder_1.html#E1:11_3_1_1_3_1_1_2"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E5:11_3_1_1_3_1_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="holder_1.html#E4:11_3_1_1_3_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="holder_1.html#E9"><i><font color="Green">Lemma7</font></i></a></i>;<br/><b>hence </b><a NAME="E6:11_3_1_1_3_1_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&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">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</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">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E4:11_3_1_1"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_1_3_1_2"><b>case </b></a><a NAME="E1:11_3_1_1_3_1_2"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:11_3_1_1_3_1_2"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="holder_1.html#E15:11"><i><font color="Green">E20</font></i></a>, <a class="txt" href="holder_1.html#E1:11_3_1_1_3_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="holder_1.html#E8"><i><font color="Green">Lemma6</font></i></a></i>;<br/><a NAME="E3:11_3_1_1_3_1_2"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1_3_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:11_3_1_1_3_1_2_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:11_3_1_1_3_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="holder_1.html#E2:11_3_1_1_3_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="holder_1.html#E1:11_3_1_1_3_1_2_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E3:11_3_1_1_3_1_2_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="holder_1.html#E13:11"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E4:11_3_1_1_3_1_2_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1_3_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_3_1_1_3_1_2_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:11_3_1_1_3_1_2_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E3:11_3_1_1_3_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>
<b>hence </b><a NAME="E4:11_3_1_1_3_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="holder_1.html#E3:11_3_1_1_3_1_2_1"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:11_3_1_1_3_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:11_3_1_1_3_1_2"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1_3_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:11_3_1_1_3_1_2_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>thus </b><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E13:11"><i><font color="Green">E18</font></i></a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E3:11_3_1_1_3_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="holder_1.html#E1:11_3_1_1_3_1_2_2"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E5:11_3_1_1_3_1_2"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="holder_1.html#E4:11_3_1_1_3_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="holder_1.html#E9"><i><font color="Green">Lemma7</font></i></a></i>;<br/><b>hence </b><a NAME="E6:11_3_1_1_3_1_2"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&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">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</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">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E4:11_3_1_1"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:11_3_1_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&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">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</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">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_2"><b>case </b></a><a NAME="E1:11_3_1_2"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:11_3_1_2"/><i><font color="Green">E24</font></i>: 
( <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="holder_1.html#E1:11_3_1_2"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E3:11_3_1_2"/><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="holder_1.html#E16:11"><i><font color="Green">E21</font></i></a>, <a class="txt" href="holder_1.html#E2:11_3_1_2"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E4:11_3_1_2"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E5:11_3_1_2"/><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="holder_1.html#E17:11"><i><font color="Green">E22</font></i></a>, <a class="txt" href="holder_1.html#E2:11_3_1_2"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E6:11_3_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</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">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E8:11_3_1_2"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</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">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E10:11_3_1_2"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i><a NAME="E11:11_3_1_2"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:11_3_1_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E2:11_3_1_2_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</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">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="holder_1.html#E4:11_3_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="real_2.html#T125">REAL_2:125</a></i>;<br/>
<b>hence </b><a NAME="E3:11_3_1_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="holder_1.html#E8:11_3_1_2"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:11_3_1_2"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:11_3_1_2_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E2:11_3_1_2_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</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">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="holder_1.html#E6:11_3_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="real_2.html#T125">REAL_2:125</a></i>;<br/>
<b>hence </b><a NAME="E3:11_3_1_2_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="holder_1.html#E10:11_3_1_2"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:11_3_1_2"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font> implies <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:11_3_1_2_3"/><i><font color="Green">E34</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E11:11_3_1_2"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E2:11_3_1_2_3"/><i><font color="Green">E35</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E12:11_3_1_2"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>thus </b><a NAME="E3:11_3_1_2_3"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font> implies <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="holder_1.html#E1:11"><i><font color="Green">E9</font></i></a>, <a class="txt" href="holder_1.html#E2:11"><i><font color="Green">E10</font></i></a>, <a class="txt" href="holder_1.html#E1:11_3_1_2_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="holder_1.html#E2:11_3_1_2_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="holder_1.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:11_3_1_2"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:11_3_1_2_4"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</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">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="holder_1.html#E8:11_3_1_2"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E2:11_3_1_2_4"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</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">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="holder_1.html#E10:11_3_1_2"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E1:11_3_1_2_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="holder_1.html#E2:11_3_1_2_4"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T77">XCMPLX_1:77</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E13:11"><i><font color="Green">E18</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E15:11_3_1_2"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:11_3_1_2_5"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</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">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="holder_1.html#E8:11_3_1_2"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E2:11_3_1_2_5"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</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">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="holder_1.html#E10:11_3_1_2"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E3:11_3_1_2_5"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E1:11_3_1_2_5"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E4:11_3_1_2_5"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_2_5_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_2_5_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_2_5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_3_1_2_5_1_1_1"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 or  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_2_5_1_1_1_1"><b>case </b></a><a NAME="E1:11_3_1_2_5_1_1_1_1"/><i><font color="Green">E40</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default">0 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="holder_1.html#E1:11_3_1_2_5_1_1_1_1"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
0 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E8:11"><i><font color="Green">E13</font></i></a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E8:11"><i><font color="Green">E13</font></i></a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E1:11_3_1_2_5_1_1_1_1"><i><font color="Green">E40</font></i></a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_2_5_1_1_1_2"><b>case </b></a><a NAME="E1:11_3_1_2_5_1_1_1_2"/><i><font color="Green">E40</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:11_3_1_2_5_1_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="holder_1.html#E1:11_3_1_2_5_1_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>hence </b><a NAME="E3:11_3_1_2_5_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E4:11_3_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="power.html#T36">POWER:36</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:11_3_1_2_5_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<i><font color="Green">E40</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="holder_1.html#E3:11_3_1_2_5"><i><font color="Green">E38</font></i></a>, <a class="txt" href="holder_1.html#E4:11_3_1_2_5"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="holder_1.html#E13:11"><i><font color="Green">E18</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="holder_1.html#E3:11_3_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="power.html#T38">POWER:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="holder_1.html#E9:11"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
;<br/>
<a NAME="E6:11_3_1_2_5"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E2:11_3_1_2_5"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E7:11_3_1_2_5"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_2_5_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_2_5_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_2_5_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_3_1_2_5_3_1_1"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 or  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_2_5_3_1_1_1"><b>case </b></a><a NAME="E1:11_3_1_2_5_3_1_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default">0 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="holder_1.html#E1:11_3_1_2_5_3_1_1_1"><i><font color="Green">E43</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="holder_1.html#E7:11"><i><font color="Green">E12</font></i></a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
<br/>.= 
0 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">0 <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E7:11"><i><font color="Green">E12</font></i></a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E1:11_3_1_2_5_3_1_1_1"><i><font color="Green">E43</font></i></a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_2_5_3_1_1_2"><b>case </b></a><a NAME="E1:11_3_1_2_5_3_1_1_2"/><i><font color="Green">E43</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:11_3_1_2_5_3_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="holder_1.html#E1:11_3_1_2_5_3_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>hence </b><a NAME="E3:11_3_1_2_5_3_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E6:11_3_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="power.html#T36">POWER:36</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:11_3_1_2_5_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<i><font color="Green">E43</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="holder_1.html#E6:11_3_1_2_5"><i><font color="Green">E41</font></i></a>, <a class="txt" href="holder_1.html#E7:11_3_1_2_5"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="holder_1.html#E13:11"><i><font color="Green">E18</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="holder_1.html#E5:11_3_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="power.html#T38">POWER:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="holder_1.html#E10:11"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
;<br/>
<a NAME="E9:11_3_1_2_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="holder_1.html#E5:11_3_1_2_5"><i><font color="Green">E40</font></i></a>, <a class="txt" href="holder_1.html#E8:11_3_1_2_5"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:11_3_1_2_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:11_3_1_2"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:11_3_1_2_6"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="holder_1.html#E14:11_3_1_2"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E2:11_3_1_2_6"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="holder_1.html#E13:11_3_1_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E3:11_3_1_2_6"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="holder_1.html#E15:11_3_1_2"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:11_3_1_2_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="holder_1.html#E1:11_3_1_2_6"><i><font color="Green">E37</font></i></a>, <a class="txt" href="holder_1.html#E2:11_3_1_2_6"><i><font color="Green">E38</font></i></a>, <a class="txt" href="holder_1.html#E3:11_3_1_2_6"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:11_3_1_2"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="holder_1.html#E16:11_3_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="series_1.html#T17">SERIES_1:17</a></i>;<br/><i><font color="Green">E38</font></i>: <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="series_1.html#T12">SERIES_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
;<br/><i><font color="Green">E39</font></i>: <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="series_1.html#T8">SERIES_1:8</a></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">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#T12">SERIES_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#T12">SERIES_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#T12">SERIES_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#T12">SERIES_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E3:11_3_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E5:11_3_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="holder_1.html#E2:11"><i><font color="Green">E10</font></i></a></i>
;<br/><a NAME="E20:11_3_1_2"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</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">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="holder_1.html#E4:11_3_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="holder_1.html#E6:11_3_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/><a NAME="E21:11_3_1_2"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="holder_1.html#E17:11_3_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="holder_1.html#E18:11_3_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="holder_1.html#E19:11_3_1_2"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E22:11_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>;<br/><b>hence </b><a NAME="E23:11_3_1_2"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&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">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</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">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E20:11_3_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="real_2.html#T118">REAL_2:118</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E19:11"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&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">Partial_Sums</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</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">Partial_Sums</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>


</div>
