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

<b>let </b><font color="Maroon">s</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">s1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:30"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> <a href="power.html#K2">-root</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">s1</font> is <a href="seq_2.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E3:30"/><i><font color="Green">E33</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/>

<b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2;<br/>
<a NAME="E4:30"/>
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E5:30"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E6:30"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T217">XREAL_1:217</a></i>;<br/>
<a NAME="E7:30"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
;<br/>
<a NAME="E8:30"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">s1</font> <a href="seqm_3.html#K1">^\</a> 1 is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> )
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:30_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E2:30_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="power.html#K2">-root</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E4:30_1"/>
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="series_1.html#T1" target="_self">Th1</a>, <a class="txt" href="series_1.html#E1:30"><i><font color="Green">E31</font></i></a>, <a class="ref" href="power.html#T8">POWER:8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:30"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="series_1.html#E1"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="prepower.html#T2">PREPOWER:2</a></i>;<br/>
<a NAME="E11:30"/><i><font color="Green">E40</font></i>: 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<a NAME="E1:30_2"/>
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="ref" href="series_1.html#T1" target="_self">Th1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:30_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="real_2.html#T109">REAL_2:109</a></i>;<br/>
<a NAME="E3:30_2"/><b>then </b>
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E4:30_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>hence </b><a NAME="E5:30_2"/>
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:30"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, , <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E14:30"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:30_3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:30_3"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:30_3"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:30_3"/><b>then </b><i><font color="Green">E45</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="series_1.html#E6:30"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:30_3"/><b>then </b><i><font color="Green">E69</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
;<br/>
<a NAME="E6:30_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E7:30_3"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="series_1.html#E2:30"><i><font color="Green">E32</font></i></a>, <a class="txt" href="series_1.html#E7:30"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E8:30_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="seq_2.html#T9">SEQ_2:9</a></i>;<br/>
<a NAME="E9:30_3"/><b>then </b>
<font color="Maroon">s1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E10:30_3"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">n</font> <a href="power.html#K2">-root</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_3_1_1"/>
( <font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <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="30_3_1_1_1"><b>suppose </b></a><a NAME="E1:30_3_1_1_1"/>
<font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_3_1_1_1"/>
<font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="series_1.html#E1:30"><i><font color="Green">E31</font></i></a>, <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="30_3_1_1_2"><b>suppose </b></a><a NAME="E1:30_3_1_1_2"/>
<font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:30_3_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="prepower.html#K5">-Root</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="series_1.html#E8:30"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="prepower.html#D3">PREPOWER:def 3</a></i>;<br/><a NAME="E3:30_3_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="power.html#K2">-root</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="series_1.html#E8:30"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="power.html#D1">POWER:def 1</a></i>;<br/><a NAME="E4:30_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="power.html#K2">-root</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="txt" href="series_1.html#E10:30"><i><font color="Green">E39</font></i></a>, <a class="ref" href="power.html#T42">POWER:42</a></i>;<br/><a NAME="E5:30_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="power.html#K2">-root</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="series_1.html#E8:30"><i><font color="Green">E36</font></i></a>, <a class="ref" href="power.html#T47">POWER:47</a></i>;<br/><b>hence </b><a NAME="E6:30_3_1_1_2"/>
<font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="series_1.html#E8:30"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="power.html#T5">POWER:5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E12:30_3"/>
<font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">s2</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="prepower.html#K2">GeoSeq</a> ;<br/>
<a NAME="E15:30"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="prepower.html#K2">GeoSeq</a>  is <a href="series_1.html#V1" target="_self">summable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4"><b>proof </b></a><div class="add">

<a NAME="E1:30_4"/>
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E2:30_4"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T217">XREAL_1:217</a></i>;<br/>
<a NAME="E3:30_4"/><b>then </b><i><font color="Green">E73</font></i>: 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K5">-</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/>
<a NAME="E4:30_4"/>
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> 0
 
<b>by </b><i><a class="ref" href="series_1.html#T1" target="_self">Th1</a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E5:30_4"/><b>then </b>
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K2">*</a> 2
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:30_4"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E7:30_4"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E8:30_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E9:30_4"/><b>then </b>
 <a href="real_1.html#K1">-</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#K5">-</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="real_1.html#K1">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E10:30_4"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="series_1.html#E7:30"><i><font color="Green">E35</font></i></a>, <a class="ref" href="seq_2.html#T9">SEQ_2:9</a></i>;<br/>
<b>hence </b><a NAME="E11:30_4"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="prepower.html#K2">GeoSeq</a>  is <a href="series_1.html#V1" target="_self">summable</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E16:30"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:30_5"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:30_5"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E3:30_5"/><b>then </b><i><font color="Green">E75</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="series_1.html#E7:30"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="prepower.html#D1">PREPOWER:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="series_1.html#E8:30"><i><font color="Green">E36</font></i></a>, <a class="ref" href="power.html#T47">POWER:47</a></i>
;<br/>
<b>hence </b><a NAME="E5:30_5"/>
<font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="series_1.html#E3:30"><i><font color="Green">E33</font></i></a>, <a class="txt" href="series_1.html#E7:30"><i><font color="Green">E35</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:30"/>
<font color="Maroon">s</font> is <a href="series_1.html#V1" target="_self">summable</a>
 <b>by </b><i>, <a class="txt" href="series_1.html#E6:30"><i><font color="Green">E34</font></i></a>, </i>;<br/>


</div>
