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

<b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:15"/><i><font color="Green">E26</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

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

<b>assume </b><b>that </b><br/><a NAME="E2:15"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">a</font> is <a href="seq_2.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E3:15"/><i><font color="Green">E28</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  0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 <b>and </b><br/><a NAME="E4:15"/><i><font color="Green">E29</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  <font color="Maroon">ap</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1"/>
(  <a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or  <a href="seq_2.html#K2">lim</a> <font color="Maroon">a</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="15_1_1_1"><b>case </b></a><a NAME="E1:15_1_1_1"/><i><font color="Green">E30</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_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="E1:15_1_1_1_1_1"/>
(  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> 0 or for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</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="15_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:15_1_1_1_1_1_1"/><i><font color="Green">E31</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">N</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="E3:15_1_1_1_1_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E4:15_1_1_1_1_1_1"/><i><font color="Green">E37</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  <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1_1_1"><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/>

<a NAME="E1:15_1_1_1_1_1_1_1"/>
<font color="Maroon">N</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E2:15_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:15_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:15_1_1_1_1_1_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>take </b><font color="Maroon">n</font> = 0;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E2:15_1_1_1_1_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><i><font color="Green">E40</font></i>: <span class="p1">(<span class="default"><font color="Maroon">ap</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> = 
<font color="Maroon">ap</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">N</font></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">a</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">N</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
0 <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
0
<b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
;<br/><a NAME="E4:15_1_1_1_1_1_1_2"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><a NAME="E5:15_1_1_1_1_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">ap</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/><b>hence </b><a NAME="E6:15_1_1_1_1_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">ap</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E6:15_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">ap</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/><a NAME="E7:15_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">ap</font> is <a href="seq_2.html#V4">convergent</a> &amp; <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">ap</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">N</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a>, <a class="ref" href="seq_4.html#T35">SEQ_4:35</a></i>;<br/><b>hence </b><a NAME="E8:15_1_1_1_1_1_1"/>
( <font color="Maroon">ap</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">ap</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:15_1_1_1_1_1_2"/><i><font color="Green">E45</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>   ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">m1</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="E3:15_1_1_1_1_1_2"/><i><font color="Green">E46</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> &amp; <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i/>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0;<br/><a NAME="E4:15_1_1_1_1_1_2"/><i><font color="Green">E47</font></i>: 
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>]
 <b>by </b><i/>;<br/><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="E6:15_1_1_1_1_1_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">M</font>] &amp; ( 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">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i><div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1_2_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/><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="E2:15_1_1_1_1_1_2_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">m</font>;<br/><b>thus </b><a NAME="E3:15_1_1_1_1_1_2_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> holds <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">m</font> &amp; <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> &amp; <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 holds <br/><font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) );<br/><a NAME="E8:15_1_1_1_1_1_2"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1_2_2"><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>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 );<br/>
<a NAME="E1:15_1_1_1_1_1_2_2"/><i><font color="Green">E52</font></i>: 
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">m</font>]
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">l</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="E3:15_1_1_1_1_1_2_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">l</font>] &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<b>take </b>
<font color="Maroon">l</font>
;<br/>

<b>thus </b><a NAME="E4:15_1_1_1_1_1_2_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Maroon">l</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:15_1_1_1_1_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>,<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>();<br/></i><a NAME="E11:15_1_1_1_1_1_2"/><i><font color="Green">E56</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E12:15_1_1_1_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/><div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1_2_3"><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="E1:15_1_1_1_1_1_2_3"/>
<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E2:15_1_1_1_1_1_2_3"/>
<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1_2_4"><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="E1:15_1_1_1_1_1_2_4"/>
( <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="txt" href="holder_1.html#E1:15"><i><font color="Green">E26</font></i></a></i>;<br/><b>hence </b><a NAME="E2:15_1_1_1_1_1_2_4"/>
<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">F</font> <a href="seq_1.html#K2">.</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/>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a> <b>by </b><i>, <a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a>, <a class="ref" href="seqm_3.html#D11">SEQM_3:def 11</a></i>;<br/><a NAME="E17:15_1_1_1_1_1_2"/><i><font color="Green">E58</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">a</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 holds <br/> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1_2_5"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<b>assume </b><a NAME="E1:15_1_1_1_1_1_2_5"/><i><font color="Green">E59</font></i>: 
 ex <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">S<sub>3</sub></font>[<font color="Olive">n</font>]
 ;<br/>

<b>consider </b><font color="Maroon">M1</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="E3:15_1_1_1_1_1_2_5"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">M1</font>] &amp; ( 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">S<sub>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">M1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="holder_1.html#E3:15"><i><font color="Green">E28</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">M1</font> &amp; <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E4:15_1_1_1_1_1_2_5"/><i><font color="Green">E61</font></i>: 
 ex <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">S<sub>4</sub></font>[<font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1_2_5_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">M</font>
;<br/>

<a NAME="E1:15_1_1_1_1_1_2_5_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M1</font>
 
<b>by </b><i>, <a class="txt" href="holder_1.html#E4:15"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E2:15_1_1_1_1_1_2_5_1"/>
<font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M1</font>
 
<b>by </b><i>, <a class="txt" href="holder_1.html#E4:15"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_1_1_1_1_2_5_1"/>
<font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">M1</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>

<b>thus </b><a NAME="E4:15_1_1_1_1_1_2_5_1"/>
<font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/>

<b>take </b>
0
;<br/>

<b>thus </b><a NAME="E5:15_1_1_1_1_1_2_5_1"/>
<font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:15_1_1_1_1_1_2_5"/><i><font color="Green">E63</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">S<sub>4</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M1</font>
 
;<br/>
<b>consider </b><font color="Maroon">MX</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="E7:15_1_1_1_1_1_2_5"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">MX</font>] &amp; ( 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">S<sub>4</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">MX</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(, <a class="txt" href="holder_1.html#E1:15_1_1_1"><i><font color="Green">E30</font></i></a>);<br/></i>
<a NAME="E8:15_1_1_1_1_1_2_5"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">MX</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">M1</font> holds <br/><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1_2_5_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:15_1_1_1_1_1_2_5_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">MX</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">M1</font> &amp; <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1_2_5_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1_2_5_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1_1_1_1_2_5_2_1_1"/>
(  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1_2_5_2_1_1_1"><b>case </b></a><a NAME="E1:15_1_1_1_1_1_2_5_2_1_1_1"/>
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:15_1_1_1_1_1_2_5_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="holder_1.html#T3" target="_self">Th3</a>, <a class="ref" href="holder_1.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1_2_5_2_1_1_2"><b>case </b></a><a NAME="E1:15_1_1_1_1_1_2_5_2_1_1_2"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:15_1_1_1_1_1_2_5_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="holder_1.html#E4:15"><i><font color="Green">E29</font></i></a>, <a class="ref" href="holder_1.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:15_1_1_1_1_1_2_5_2"/>
contradiction
 ;<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="E10:15_1_1_1_1_1_2_5"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MX</font>
 <b>by </b><i><a class="ref" href="holder_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E11:15_1_1_1_1_1_2_5"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">MX</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">MX</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> &amp; <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 holds <br/><font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) )
 
<b>by </b><i>, <a class="ref" href="holder_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E12:15_1_1_1_1_1_2_5"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M1</font>
 
<b>by </b><i>, <a class="txt" href="holder_1.html#E4:15"><i><font color="Green">E29</font></i></a>, <a class="ref" href="holder_1.html#T3" target="_self">Th3</a>, <a class="ref" href="holder_1.html#T4" target="_self">Th4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1_2_5_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1_1_1_1_2_5_3"/>
<font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M1</font>
 ;<br/><a NAME="E2:15_1_1_1_1_1_2_5_3"/><b>then </b>
<font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">M1</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:15_1_1_1_1_1_2_5_3"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:15_1_1_1_1_1_2_5"/>
contradiction
 <b>by </b><i><a class="txt" href="holder_1.html#E4:15"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:15_1_1_1_1_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">a</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/><a NAME="E19:15_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">a</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, <a class="ref" href="seq_4.html#T29">SEQ_4:29</a></i>;<br/><a NAME="E20:15_1_1_1_1_1_2"/><i><font color="Green">E74</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="txt" href="holder_1.html#E3:15"><i><font color="Green">E28</font></i></a>, <a class="ref" href="seq_4.html#T30">SEQ_4:30</a></i>;<br/><a NAME="E21:15_1_1_1_1_1_2"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1_2_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:15_1_1_1_1_1_2_6"/><i><font color="Green">E76</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 ;<br/><a NAME="E2:15_1_1_1_1_1_2_6"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><b>then consider </b><font color="Maroon">n</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="E4:15_1_1_1_1_1_2_6"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E4:15"><i><font color="Green">E29</font></i></a>, <a class="txt" href="holder_1.html#E1:15_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/><b>take </b><font color="Maroon">k</font> = <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E5:15_1_1_1_1_1_2_6"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1_2_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1_2_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1_1_1_1_2_6_1_1"/>
( <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</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="15_1_1_1_1_1_2_6_1_1_1"><b>case </b></a><a NAME="E1:15_1_1_1_1_1_2_6_1_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><font color="Maroon">ap</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> = 
0 <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
<b>by </b><i>, </i>
<br/>.= 
0
<b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
;<br/><a NAME="E3:15_1_1_1_1_1_2_6_1_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ap</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/><b>hence </b><a NAME="E4:15_1_1_1_1_1_2_6_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ap</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="holder_1.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1_2_6_1_1_2"><b>case </b></a><a NAME="E1:15_1_1_1_1_1_2_6_1_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:15_1_1_1_1_1_2_6_1_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">l</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="E4:15_1_1_1_1_1_2_6_1_1_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E2:15"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E5:15_1_1_1_1_1_2_6_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="holder_1.html#T4" target="_self">Th4</a>, , <a class="ref" href="seqm_3.html#T7">SEQM_3:7</a></i>;<br/><a NAME="E6:15_1_1_1_1_1_2_6_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">F</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">l</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:15_1_1_1_1_1_2_6_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="seqm_3.html#T31">SEQM_3:31</a></i>;<br/><a NAME="E8:15_1_1_1_1_1_2_6_1_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E9:15_1_1_1_1_1_2_6_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/><a NAME="E10:15_1_1_1_1_1_2_6_1_1_2"/><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">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="power.html#T42">POWER:42</a></i>;<br/><a NAME="E11:15_1_1_1_1_1_2_6_1_1_2"/><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">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="holder_1.html#T3" target="_self">Th3</a>, <a class="ref" href="power.html#T38">POWER:38</a></i>;<br/><a NAME="E12:15_1_1_1_1_1_2_6_1_1_2"/><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">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="power.html#K3">to_power</a> 1
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>;<br/><a NAME="E13:15_1_1_1_1_1_2_6_1_1_2"/><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">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/><a NAME="E14:15_1_1_1_1_1_2_6_1_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E15:15_1_1_1_1_1_2_6_1_1_2"/><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ap</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><a NAME="E16:15_1_1_1_1_1_2_6_1_1_2"/><i><font color="Green">E86</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ap</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E17:15_1_1_1_1_1_2_6_1_1_2"/>
<font color="Maroon">ap</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, </i>;<br/><a NAME="E18:15_1_1_1_1_1_2_6_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">ap</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E19:15_1_1_1_1_1_2_6_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ap</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:15_1_1_1_1_1_2_6"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ap</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div><a NAME="E23:15_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
for <font color="Olive">e</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</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">ap</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e</font>
 ;<br/><b>hence </b><a NAME="E24:15_1_1_1_1_1_2"/>
<font color="Maroon">ap</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/><b>hence </b><a NAME="E25:15_1_1_1_1_1_2"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">ap</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="holder_1.html#T3" target="_self">Th3</a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:15_1_1_1"/>
( <font color="Maroon">ap</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">ap</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2"><b>case </b></a><a NAME="E1:15_1_1_2"/><i><font color="Green">E89</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:15_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="seq_2.html#T31">SEQ_2:31</a></i>;<br/><a NAME="E3:15_1_1_2"/><b>then </b><i><font color="Green">E91</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E4:15_1_1_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Olive">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">e0</font> =  <a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font>;<br/>
<a NAME="E1:15_1_1_2_1"/><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></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="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<b>consider </b><font color="Maroon">k</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="E3:15_1_1_2_1"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</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">a</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</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">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></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/>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

<div><i><font color="Green">E95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:15_1_1_2_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E2:15_1_1_2_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></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">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i/>;<br/><a NAME="E3:15_1_1_2_1_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></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">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/><a NAME="E4:15_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></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">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></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">a</font></span>)</span></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">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E5:15_1_1_2_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:15_1_1_2_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:15_1_1_2_1_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">n</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="E3:15_1_1_2_1_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/><a NAME="E4:15_1_1_2_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E5:15_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g</font> </span>}</span> 
 ;<br/><a NAME="E6:15_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/><a NAME="E7:15_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="taylor_1.html#D4">TAYLOR_1:def 4</a></i>;<br/><b>hence </b><a NAME="E8:15_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:15_1_1_2_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">k</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="E6:15_1_1_2"/><i><font color="Green">E99</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span>
 ;<br/><a NAME="E7:15_1_1_2"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font> 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">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>;<br/><a NAME="E8:15_1_1_2"/><i><font color="Green">E101</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E9:15_1_1_2"/>
 <a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font> <a href="fdiff_1.html#R1">is_differentiable_in</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="taylor_1.html#T21">TAYLOR_1:21</a></i>;<br/><a NAME="E10:15_1_1_2"/><b>then </b>
 <a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font> <a href="fcont_1.html#R1">is_continuous_in</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="fdiff_1.html#T32">FDIFF_1:32</a></i>;<br/><a NAME="E11:15_1_1_2"/><b>then </b><i><font color="Green">E102</font></i>: 
( <span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp; <span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="fcont_1.html#D1">FCONT_1:def 1</a></i>;<br/><a NAME="E12:15_1_1_2"/><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">ap</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:15_1_1_2_2_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E3:15_1_1_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10">RFUNCT_2:10</a></i>;<br/><a NAME="E4:15_1_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:15_1_1_2_2_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <b>by </b><i><a class="ref" href="taylor_1.html#D4">TAYLOR_1:def 4</a></i>;<br/><a NAME="E6:15_1_1_2_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/><a NAME="E7:15_1_1_2_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">g</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:15_1_1_2_2_1"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><a NAME="E10:15_1_1_2_2_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</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="holder_1.html#E1:15"><i><font color="Green">E26</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="taylor_1.html#D4">TAYLOR_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font>
<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">a</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
<b>by </b><i><a class="txt" href="holder_1.html#E2:15"><i><font color="Green">E27</font></i></a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
<br/>.= 
<font color="Maroon">ap</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ap</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:15_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span> <a href="rfunct_2.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">ap</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T8">SEQ_1:8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:15_1_1_2"/><i><font color="Green">E108</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_1_2_3"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:15_1_1_2_3"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green">E109</font></i>: <span class="p1">(<span class="default"><a href="taylor_1.html#K4">#R</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="prepower.html#K13">#R</a> <font color="Maroon">p</font>
<b>by </b><i>, , <a class="ref" href="taylor_1.html#D4">TAYLOR_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
;<br/><a NAME="E15:15_1_1_2"/>
( <font color="Maroon">ap</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font> 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">ap</font> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E16:15_1_1_2"/>
( <font color="Maroon">ap</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">ap</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="seq_4.html#T35">SEQ_4:35</a>, <a class="ref" href="seq_4.html#T36">SEQ_4:36</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:15"/>
( <font color="Maroon">ap</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">ap</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">a</font></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">p</font> )
 ;<br/>


</div>
