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

<b>let </b><font color="Maroon" title="c1">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">s1</font>, <font color="Maroon" title="c3">s2</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c2">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green" title="E19">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c3">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c1">m</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">s</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> $1 ) holds <br/>( <font color="Olive" title="b1">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> $1 );<br/>
<a NAME="E3:23"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">s</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green" title="E21">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> 0
 ;<br/>

<div><i><font color="Green" title="E22">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>thus </b><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> 0
<b>by </b><i><a class="txt" href="prepower.html#E1:23_1"><i><font color="Green" title="E21">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
<b>by </b><i><a class="ref" href="prepower.html#D1" target="_self" title="PREPOWER:def.1">Def1</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="prepower.html#T4" target="_self" title="PREPOWER:th.4">Th4</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E3:23_1"/><b>then </b><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c4">s</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D6" title="SEQM_3:def.6">SEQM_3:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:23_1"/>
<font color="Maroon" title="c4">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T39" title="SEQ_4:th.39">SEQ_4:39</a></i>;<br/>

<b>thus </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s</font> = 
<font color="Maroon" title="c4">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
<b>by </b><i><a class="txt" href="prepower.html#E3:23_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="seq_4.html#T41" title="SEQ_4:th.41">SEQ_4:41</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="prepower.html#E2:23_1"><i><font color="Green" title="E22">A5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
<b>by </b><i><a class="ref" href="prepower.html#T4" target="_self" title="PREPOWER:th.4">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> 0
<b>by </b><i><a class="ref" href="prepower.html#D1" target="_self" title="PREPOWER:def.1">Def1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:23"/><i><font color="Green" title="E21">A7</font></i>: 
for <font color="Olive" title="b1">m1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m1</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">m1</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:23_2"/><i><font color="Green" title="E22">A8</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">m1</font> ) holds <br/>( <font color="Olive" title="b1">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">m1</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c5">s</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E2:23_2"/><i><font color="Green" title="E23">A9</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c5">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1</span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">m1</font>;<br/>
<b>consider </b><font color="Maroon" title="c6">s3</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E4:23_2"/><i><font color="Green" title="E24">A10</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c6">s3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E5:23_2"/><i><font color="Green" title="E25">A11</font></i>: 
( <font color="Maroon" title="c6">s3</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">m1</font> )
 
<b>by </b><i><a class="txt" href="prepower.html#E1:23_2"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="prepower.html#E4:23_2"><i><font color="Green" title="E24">A10</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>thus </b><font color="Maroon" title="c5">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E2:23_2"><i><font color="Green" title="E23">A9</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">m1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E4:23_2"><i><font color="Green" title="E24">A10</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E7:23_2"/><b>then </b><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s3</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <font color="Maroon" title="c2">s1</font>
 
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>;<br/>
<b>hence </b><a NAME="E8:23_2"/>
<font color="Maroon" title="c5">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="prepower.html#E1:23"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="prepower.html#E5:23_2"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></i>;<br/>

<b>thus </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c5">s</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c4">m1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:23"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="prepower.html#E5:23_2"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="prepower.html#E7:23_2"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:23"/>
for <font color="Olive" title="b1">m1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m1</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="prepower.html#E3:23"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="prepower.html#E4:23"><i><font color="Green" title="E21">A7</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:23"/>
( <font color="Maroon" title="c3">s2</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K3" title="PREPOWER:func.3">|^</a> <font color="Maroon" title="c1">m</font> )
 <b>by </b><i><a class="txt" href="prepower.html#E2:23"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>


</div>
