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

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

<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E13">A1</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">p</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">ap</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="E2:15"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c2">a</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>and </b><br/><a NAME="E3:15"/><i><font color="Green" title="E15">A3</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   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font>
 <b>and </b><br/><a NAME="E4:15"/><i><font color="Green" title="E16">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="c3">ap</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</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">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">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" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </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" title="E17">A5</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><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" 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>  st <br/>for <font color="Olive" title="b2">m</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="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or 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>   ex <font color="Olive" title="b2">m</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 <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 </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"/>
 ex <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>  st <br/>for <font color="Olive" title="b2">m</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="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">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> <b> such that </b><br/><a NAME="E3:15_1_1_1_1_1_1"/><i><font color="Green" title="E18">A6</font></i>: 
for <font color="Olive" title="b1">m</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" title="c4">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E4:15_1_1_1_1_1_1"/><i><font color="Green" title="E19">A7</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <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" 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/>

<a NAME="E1:15_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">N</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E2:15_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="holder_1.html#E3:15_1_1_1_1_1_1"><i><font color="Green" title="E18">A6</font></i></a></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" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green" title="E20">A8</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" title="c5">e</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:15_1_1_1_1_1_1_2"/><i><font color="Green" title="E21">A9</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>take </b><font color="Maroon" title="c6">n</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><b>let </b><font color="Maroon" title="c7">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>assume </b><a NAME="E2:15_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font>
 ;<br/><i><font color="Green" title="E22">A10</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">ap</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">m</font> = 
<font color="Maroon" title="c3">ap</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">N</font></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">N</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="txt" href="holder_1.html#E4:15"><i><font color="Green" title="E16">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="txt" href="holder_1.html#E4:15_1_1_1_1_1_1"><i><font color="Green" title="E19">A7</font></i></a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="holder_1.html#E1:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></i>
;<br/><a NAME="E4:15_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="holder_1.html#E1:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="holder_1.html#E1:15_1_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></i>;<br/><b>hence </b><a NAME="E5:15_1_1_1_1_1_1_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">ap</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E1:15_1_1_1_1_1_1_2"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="holder_1.html#E3:15_1_1_1_1_1_1_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>;<br/></div><b>end;</b></div><a NAME="E6:15_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">ap</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">N</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a></i>;<br/><a NAME="E7:15_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c3">ap</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ap</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">N</font></span>)</span> )
 <b>by </b><i><a class="txt" href="holder_1.html#E5:15_1_1_1_1_1_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a>, <a class="ref" href="seq_4.html#T35" title="SEQ_4:th.35">SEQ_4:35</a></i>;<br/><b>hence </b><a NAME="E8:15_1_1_1_1_1_1"/>
( <font color="Maroon" title="c3">ap</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">ap</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">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> )
 <b>by </b><i><a class="ref" href="seq_4.html#T33" title="SEQ_4:th.33">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" title="E18">A11</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>   ex <font color="Olive" title="b2">m</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 <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">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> <b> such that </b><br/><a NAME="E3:15_1_1_1_1_1_2"/><i><font color="Green" title="E19">A12</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m1</font> &amp; <font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c4">m1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><a NAME="E4:15_1_1_1_1_1_2"/><i><font color="Green" title="E20">A13</font></i>: 
 ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 <b>by </b><i><a class="txt" href="holder_1.html#E3:15_1_1_1_1_1_2"><i><font color="Green" title="E19">A12</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c5">M</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:15_1_1_1_1_1_2"/><i><font color="Green" title="E21">A14</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">M</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c5">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="holder_1.html#E4:15_1_1_1_1_1_2"><i><font color="Green" title="E20">A13</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c6">M</font> = <font color="Maroon" title="c5">M</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><div><i><font color="Green" title="E22">A15</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" 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>consider </b><font color="Maroon" title="c8">m</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> <b> such that </b><br/><a NAME="E2:15_1_1_1_1_1_2_1"/><i><font color="Green" title="E23">A16</font></i>: 
( <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font> &amp; <font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="holder_1.html#E1:15_1_1_1_1_1_2"><i><font color="Green" title="E18">A11</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c9">m</font> = <font color="Maroon" title="c8">m</font>;<br/><b>thus </b><a NAME="E3:15_1_1_1_1_1_2_1"/>
( <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="holder_1.html#E2:15_1_1_1_1_1_2_1"><i><font color="Green" title="E23">A16</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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 $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> holds <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b3">k</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="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">k</font> &amp; <font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> ) );<br/><a NAME="E9:15_1_1_1_1_1_2"/><i><font color="Green" title="E23">A17</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">x</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>   ex <font color="Olive" title="b3">y</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>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">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" title="c7">n</font>, <font color="Maroon" title="c8">x</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>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> $1 &amp; <font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  );<br/>
<a NAME="E1:15_1_1_1_1_1_2_2"/>
 ex <font color="Olive" title="b1">m</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>3</sub></font>[<font color="Olive" title="b1">m</font>]
 
<b>by </b><i><a class="txt" href="holder_1.html#E8:15_1_1_1_1_1_2"><i><font color="Green" title="E22">A15</font></i></a></i>;<br/>
<a NAME="E2:15_1_1_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E24">A18</font></i>: 
 ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">m</font>]
 
<b>by </b><i><a class="txt" href="holder_1.html#E8:15_1_1_1_1_1_2"><i><font color="Green" title="E22">A15</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:15_1_1_1_1_1_2_2"/><i><font color="Green" title="E25">A19</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c9">l</font>] &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon" title="c9">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="holder_1.html#E2:15_1_1_1_1_1_2_2"><i><font color="Green" title="E24">A18</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c9">l</font>
;<br/>

<a NAME="E5:15_1_1_1_1_1_2_2"/>
<font color="Maroon" title="c9">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E6:15_1_1_1_1_1_2_2"/>
( <font color="Maroon" title="c9">l</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c9">l</font> is    <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>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">l</font>] )
 <b>by </b><i><a class="txt" href="holder_1.html#E4:15_1_1_1_1_1_2_2"><i><font color="Green" title="E25">A19</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c7">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E11:15_1_1_1_1_1_2"/><i><font color="Green" title="E24">A20</font></i>: 
( <font color="Maroon" title="c7">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font> &amp; ( 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">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c7">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c7">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="holder_1.html#E9:15_1_1_1_1_1_2"><i><font color="Green" title="E23">A17</font></i></a>);<br/></i><a NAME="E12:15_1_1_1_1_1_2"/><i><font color="Green" title="E25">A21</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><a NAME="E13:15_1_1_1_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">F</font> = <font color="Maroon" title="c7">F</font> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <b>by </b><i><a class="txt" href="holder_1.html#E12:15_1_1_1_1_1_2"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/><div><i><font color="Green" title="E26">A22</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" title="c9">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:15_1_1_1_1_1_2_3"/>
<font color="Maroon" title="c8">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">F</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E12:15_1_1_1_1_1_2"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E2:15_1_1_1_1_1_2_3"/>
<font color="Maroon" title="c8">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font> is    <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>by </b><i><a class="txt" href="holder_1.html#E12:15_1_1_1_1_1_2"><i><font color="Green" title="E25">A21</font></i></a></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" title="c9">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:15_1_1_1_1_1_2_4"/>
( <font color="Maroon" title="c8">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font> is    <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>  &amp; <font color="Maroon" title="c8">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is    <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>by </b><i><a class="txt" href="holder_1.html#E15:15_1_1_1_1_1_2"><i><font color="Green" title="E26">A22</font></i></a></i>;<br/><b>hence </b><a NAME="E2:15_1_1_1_1_1_2_4"/>
<font color="Maroon" title="c8">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E11:15_1_1_1_1_1_2"><i><font color="Green" title="E24">A20</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c9">F</font> = <font color="Maroon" title="c8">F</font> as  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a> <b>by </b><i><a class="txt" href="holder_1.html#E12:15_1_1_1_1_1_2"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="seqm_3.html#D8" title="SEQM_3:def.8">SEQM_3:def 8</a>, <a class="ref" href="seqm_3.html#D11" title="SEQM_3:def.11">SEQM_3:def 11</a></i>;<br/><a NAME="E18:15_1_1_1_1_1_2"/><i><font color="Green" title="E27">A23</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>   st <font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> ex <font color="Olive" title="b2">m</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" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( <font color="Maroon" title="c2">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> $1 ) );<br/>
<b>assume </b><a NAME="E1:15_1_1_1_1_1_2_5"/>
 ex <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>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 ;<br/>

<a NAME="E2:15_1_1_1_1_1_2_5"/><b>then </b><i><font color="Green" title="E28">A24</font></i>: 
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c10">M1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:15_1_1_1_1_1_2_5"/><i><font color="Green" title="E29">A25</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">M1</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c10">M1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="holder_1.html#E2:15_1_1_1_1_1_2_5"><i><font color="Green" title="E28">A24</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">M1</font> &amp; <font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  ex <font color="Olive" title="b1">m</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" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 );<br/>
<a NAME="E5:15_1_1_1_1_1_2_5"/><i><font color="Green" title="E30">A26</font></i>: 
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">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" title="c6">M</font>
;<br/>

<a NAME="E1:15_1_1_1_1_1_2_5_1"/><i><font color="Green" title="E31">A27</font></i>: 
<font color="Maroon" title="c6">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">M1</font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E6:15_1_1_1_1_1_2"><i><font color="Green" title="E21">A14</font></i></a>, <a class="txt" href="holder_1.html#E4:15_1_1_1_1_1_2_5"><i><font color="Green" title="E29">A25</font></i></a></i>;<br/>
<a NAME="E2:15_1_1_1_1_1_2_5_1"/>
<font color="Maroon" title="c6">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">M1</font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E11:15_1_1_1_1_1_2"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="holder_1.html#E4:15_1_1_1_1_1_2_5"><i><font color="Green" title="E29">A25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_1_1_1_1_2_5_1"/>
<font color="Maroon" title="c6">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">M1</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E1:15_1_1_1_1_1_2_5_1"><i><font color="Green" title="E31">A27</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="holder_1.html#E6:15_1_1_1_1_1_2"><i><font color="Green" title="E21">A14</font></i></a></i>;<br/>

<b>take </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
;<br/>

<b>thus </b><a NAME="E5:15_1_1_1_1_1_2_5_1"/>
<font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E11:15_1_1_1_1_1_2"><i><font color="Green" title="E24">A20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:15_1_1_1_1_1_2_5"/><i><font color="Green" title="E31">A28</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">M1</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c11">MX</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E8:15_1_1_1_1_1_2_5"/><i><font color="Green" title="E32">A29</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c11">MX</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">MX</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="holder_1.html#E6:15_1_1_1_1_1_2_5"><i><font color="Green" title="E31">A28</font></i></a>, <a class="txt" href="holder_1.html#E5:15_1_1_1_1_1_2_5"><i><font color="Green" title="E30">A26</font></i></a>);<br/></i>
<a NAME="E9:15_1_1_1_1_1_2_5"/><i><font color="Green" title="E33">A30</font></i>: 
for <font color="Olive" title="b1">k</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" title="c11">MX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">M1</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<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" title="c12">k</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> <b> such that </b><a NAME="E2:15_1_1_1_1_1_2_5_2"/><i><font color="Green" title="E34">A31</font></i>: 
( <font color="Maroon" title="c11">MX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">k</font> &amp; <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">M1</font> &amp; <font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<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" title="b1">m</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" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k</font> or for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">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" title="b1">m</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" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="txt" href="holder_1.html#E8:15_1_1_1_1_1_2_5"><i><font color="Green" title="E32">A29</font></i></a>, <a class="txt" href="holder_1.html#E2:15_1_1_1_1_1_2_5_2"><i><font color="Green" title="E34">A31</font></i></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" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">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_1_1_1_1_1_2_5"><i><font color="Green" title="E29">A25</font></i></a>, <a class="txt" href="holder_1.html#E2:15_1_1_1_1_1_2_5_2"><i><font color="Green" title="E34">A31</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:15_1_1_1_1_1_2_5_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c12">m</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> <b> such that </b><br/><a NAME="E11:15_1_1_1_1_1_2_5"/><i><font color="Green" title="E34">A32</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">MX</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E8:15_1_1_1_1_1_2_5"><i><font color="Green" title="E32">A29</font></i></a></i>;<br/>
<a NAME="E12:15_1_1_1_1_1_2_5"/><i><font color="Green" title="E35">A33</font></i>: 
( <font color="Maroon" title="c11">MX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">k</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" title="c11">MX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) )
 
<b>by </b><i><a class="txt" href="holder_1.html#E11:15_1_1_1_1_1_2"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="holder_1.html#E11:15_1_1_1_1_1_2_5"><i><font color="Green" title="E34">A32</font></i></a></i>;<br/>
<a NAME="E13:15_1_1_1_1_1_2_5"/>
<font color="Maroon" title="c10">M1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E14:15_1_1_1_1_1_2_5"/><b>then </b><i><font color="Green" title="E36">A34</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">M1</font>
 
<b>by </b><i><a class="txt" href="holder_1.html#E11:15_1_1_1_1_1_2"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="holder_1.html#E4:15_1_1_1_1_1_2_5"><i><font color="Green" title="E29">A25</font></i></a>, <a class="txt" href="holder_1.html#E8:15_1_1_1_1_1_2_5"><i><font color="Green" title="E32">A29</font></i></a>, <a class="txt" href="holder_1.html#E11:15_1_1_1_1_1_2_5"><i><font color="Green" title="E34">A32</font></i></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" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">M1</font>
 ;<br/><a NAME="E2:15_1_1_1_1_1_2_5_3"/><b>then </b>
<font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">M1</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E14:15_1_1_1_1_1_2_5"><i><font color="Green" title="E36">A34</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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><a class="txt" href="holder_1.html#E9:15_1_1_1_1_1_2_5"><i><font color="Green" title="E33">A30</font></i></a>, <a class="txt" href="holder_1.html#E12:15_1_1_1_1_1_2_5"><i><font color="Green" title="E35">A33</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:15_1_1_1_1_1_2_5"/>
contradiction
 <b>by </b><i><a class="txt" href="holder_1.html#E4:15_1_1_1_1_1_2_5"><i><font color="Green" title="E29">A25</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:15_1_1_1_1_1_2"/><i><font color="Green" title="E28">A35</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font> is    <a href="seqm_3.html#M1" title="SEQM_3:mode.1">subsequence</a> of <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D10" title="SEQM_3:def.10">SEQM_3:def 10</a></i>;<br/><a NAME="E20:15_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E29">A36</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="holder_1.html#E2:15"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="seq_4.html#T29" title="SEQ_4:th.29">SEQ_4:29</a></i>;<br/><a NAME="E21:15_1_1_1_1_1_2"/><i><font color="Green" title="E30">A37</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="holder_1.html#E2:15"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="holder_1.html#E1:15_1_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="holder_1.html#E19:15_1_1_1_1_1_2"><i><font color="Green" title="E28">A35</font></i></a>, <a class="ref" href="seq_4.html#T30" title="SEQ_4:th.30">SEQ_4:30</a></i>;<br/><a NAME="E22:15_1_1_1_1_1_2"/><i><font color="Green" title="E31">A38</font></i>: 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="holder_1.html#E1:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="holder_1.html#E1:15_1_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></i>;<br/><div><i><font color="Green" title="E32">A39</font></i>: <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" title="c10">e</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:15_1_1_1_1_1_2_6"/><i><font color="Green" title="E33">A40</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">e</font>
 ;<br/><a NAME="E2:15_1_1_1_1_1_2_6"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">e</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">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> <b> such that </b><br/><a NAME="E4:15_1_1_1_1_1_2_6"/><i><font color="Green" title="E34">A41</font></i>: 
for <font color="Olive" title="b1">m</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" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">e</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E20:15_1_1_1_1_1_2"><i><font color="Green" title="E29">A36</font></i></a>, <a class="txt" href="holder_1.html#E21:15_1_1_1_1_1_2"><i><font color="Green" title="E30">A37</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/><b>take </b><font color="Maroon" title="c12">k</font> = <font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c11">n</font>;<br/><b>let </b><font color="Maroon" title="c13">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>assume </b><a NAME="E5:15_1_1_1_1_1_2_6"/><i><font color="Green" title="E35">A42</font></i>: 
<font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">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" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </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"/>
<font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>then </b><font color="Maroon" title="c3">ap</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c13">m</font> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="txt" href="holder_1.html#E4:15"><i><font color="Green" title="E16">A4</font></i></a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="holder_1.html#E1:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></i>
;<br/><b>hence </b><a NAME="E3:15_1_1_1_1_1_2_6_1_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">ap</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">e</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E22:15_1_1_1_1_1_2"><i><font color="Green" title="E31">A38</font></i></a>, <a class="txt" href="holder_1.html#E1:15_1_1_1_1_1_2_6"><i><font color="Green" title="E33">A40</font></i></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</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" title="E36">A43</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:15_1_1_1_1_1_2_6_1_1_2"/><b>then </b><i><font color="Green" title="E37">A44</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="holder_1.html#E3:15"><i><font color="Green" title="E15">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c14">l</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> <b> such that </b><br/><a NAME="E4:15_1_1_1_1_1_2_6_1_1_2"/><i><font color="Green" title="E38">A45</font></i>: 
<font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c14">l</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E18:15_1_1_1_1_1_2"><i><font color="Green" title="E27">A23</font></i></a>, <a class="txt" href="holder_1.html#E1:15_1_1_1_1_1_2_6_1_1_2"><i><font color="Green" title="E36">A43</font></i></a></i>;<br/><a NAME="E5:15_1_1_1_1_1_2_6_1_1_2"/>
<font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">l</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E5:15_1_1_1_1_1_2_6"><i><font color="Green" title="E35">A42</font></i></a>, <a class="txt" href="holder_1.html#E4:15_1_1_1_1_1_2_6_1_1_2"><i><font color="Green" title="E38">A45</font></i></a>, <a class="ref" href="seqm_3.html#T7" title="SEQM_3:th.7">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" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c14">l</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">e</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E4:15_1_1_1_1_1_2_6"><i><font color="Green" title="E34">A41</font></i></a></i>;<br/><a NAME="E7:15_1_1_1_1_1_2_6_1_1_2"/><b>then </b><i><font color="Green" title="E39">A46</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c14">l</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">e</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="seqm_3.html#T31" title="SEQM_3:th.31">SEQM_3:31</a></i>;<br/><a NAME="E8:15_1_1_1_1_1_2_6_1_1_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="holder_1.html#E1:15_1_1_1_1_1_2_6_1_1_2"><i><font color="Green" title="E36">A43</font></i></a>, <a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></i>;<br/><a NAME="E9:15_1_1_1_1_1_2_6_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">e</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E1:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="holder_1.html#E4:15_1_1_1_1_1_2_6_1_1_2"><i><font color="Green" title="E38">A45</font></i></a>, <a class="txt" href="holder_1.html#E7:15_1_1_1_1_1_2_6_1_1_2"><i><font color="Green" title="E39">A46</font></i></a>, <a class="ref" href="power.html#T42" title="POWER:th.42">POWER:42</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" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">e</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E1:15_1_1_1_1_1_2_6"><i><font color="Green" title="E33">A40</font></i></a>, <a class="ref" href="power.html#T38" title="POWER:th.38">POWER:38</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" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">e</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> 1
 <b>by </b><i><a class="txt" href="holder_1.html#E1:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T107" title="XCMPLX_1:th.107">XCMPLX_1:107</a></i>;<br/><a NAME="E12:15_1_1_1_1_1_2_6_1_1_2"/><b>then </b><i><font color="Green" title="E40">A47</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">e</font>
 <b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>;<br/><a NAME="E13:15_1_1_1_1_1_2_6_1_1_2"/><i><font color="Green" title="E41">A48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ap</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c13">m</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E4:15"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/><a NAME="E14:15_1_1_1_1_1_2_6_1_1_2"/><b>then </b><i><font color="Green" title="E42">A49</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">ap</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c13">m</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E2:15_1_1_1_1_1_2_6_1_1_2"><i><font color="Green" title="E37">A44</font></i></a>, <a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>;<br/><a NAME="E15:15_1_1_1_1_1_2_6_1_1_2"/>
<font color="Maroon" title="c3">ap</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">e</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E2:15_1_1_1_1_1_2_6_1_1_2"><i><font color="Green" title="E37">A44</font></i></a>, <a class="txt" href="holder_1.html#E12:15_1_1_1_1_1_2_6_1_1_2"><i><font color="Green" title="E40">A47</font></i></a>, <a class="txt" href="holder_1.html#E13:15_1_1_1_1_1_2_6_1_1_2"><i><font color="Green" title="E41">A48</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E16:15_1_1_1_1_1_2_6_1_1_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">ap</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">e</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E22:15_1_1_1_1_1_2"><i><font color="Green" title="E31">A38</font></i></a>, <a class="txt" href="holder_1.html#E14:15_1_1_1_1_1_2_6_1_1_2"><i><font color="Green" title="E42">A49</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></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" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">ap</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">e</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E24:15_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">ap</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">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" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">ap</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">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E23:15_1_1_1_1_1_2"><i><font color="Green" title="E32">A39</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">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" title="c3">ap</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">ap</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">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">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" title="E17">A50</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:15_1_1_2"/><i><font color="Green" title="E18">A51</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E2:15"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="holder_1.html#E3:15"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="seq_2.html#T31" title="SEQ_2:th.31">SEQ_2:31</a></i>;<br/><a NAME="E3:15_1_1_2"/><i><font color="Green" title="E19">A52</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="holder_1.html#E2:15"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="holder_1.html#E3:15"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="holder_1.html#E1:15_1_1_2"><i><font color="Green" title="E17">A50</font></i></a>, <a class="ref" href="seq_2.html#T31" title="SEQ_2:th.31">SEQ_2:31</a></i>;<br/><a NAME="E4:15_1_1_2"/>
 ex <font color="Olive" title="b1">k</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  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c1">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" title="c4">e0</font> =  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font>;<br/>
<a NAME="E1:15_1_1_2_1"/><i><font color="Green" title="E20">A53</font></i>: 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="holder_1.html#E1:15_1_1_2"><i><font color="Green" title="E17">A50</font></i></a>, <a class="txt" href="holder_1.html#E2:15_1_1_2"><i><font color="Green" title="E18">A51</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">k</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> <b> such that </b><br/><a NAME="E3:15_1_1_2_1"/><i><font color="Green" title="E21">A54</font></i>: 
for <font color="Olive" title="b1">m</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" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</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">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="txt" href="holder_1.html#E2:15"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">k</font>
;<br/>

<div><i><font color="Green" title="E22">A55</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" title="c6">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/><a NAME="E1:15_1_1_2_1_1"/>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E2:15_1_1_2_1_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</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">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="txt" href="holder_1.html#E3:15_1_1_2_1"><i><font color="Green" title="E21">A54</font></i></a></i>;<br/><a NAME="E3:15_1_1_2_1_1"/><b>then </b>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <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">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">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" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E5:15_1_1_2_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E1:15_1_1_2_1"><i><font color="Green" title="E20">A53</font></i></a>, <a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">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" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:15_1_1_2_1_2"/><i><font color="Green" title="E23">A56</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c5">k</font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon" title="c7">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> <b> such that </b><br/><a NAME="E3:15_1_1_2_1_2"/><i><font color="Green" title="E24">A57</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E1:15_1_1_2_1_2"><i><font color="Green" title="E23">A56</font></i></a>, <a class="ref" href="rfunct_2.html#T9" title="RFUNCT_2:th.9">RFUNCT_2:9</a></i>;<br/><a NAME="E4:15_1_1_2_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E4:15_1_1_2_1"><i><font color="Green" title="E22">A55</font></i></a></i>;<br/><a NAME="E5:15_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">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" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="limfunc1.html#T152" title="LIMFUNC1:th.152">LIMFUNC1:152</a></i>;<br/><b>hence </b><a NAME="E7:15_1_1_2_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E3:15_1_1_2_1_2"><i><font color="Green" title="E24">A57</font></i></a>, <a class="ref" href="taylor_1.html#D4" title="TAYLOR_1:def.4">TAYLOR_1:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:15_1_1_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c4">k</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> <b> such that </b><br/><a NAME="E6:15_1_1_2"/><i><font color="Green" title="E20">A58</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c1">p</font></span>)</span>
 ;<br/><a NAME="E7:15_1_1_2"/><i><font color="Green" title="E21">A59</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="txt" href="holder_1.html#E2:15"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="seq_4.html#T33" title="SEQ_4:th.33">SEQ_4:33</a></i>;<br/><a NAME="E8:15_1_1_2"/><b>then </b>
 <a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c1">p</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E3:15_1_1_2"><i><font color="Green" title="E19">A52</font></i></a>, <a class="ref" href="taylor_1.html#T21" title="TAYLOR_1:th.21">TAYLOR_1:21</a></i>;<br/><a NAME="E9:15_1_1_2"/><b>then </b>
 <a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c1">p</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="fdiff_1.html#T32" title="FDIFF_1:th.32">FDIFF_1:32</a></i>;<br/><a NAME="E10:15_1_1_2"/><b>then </b><i><font color="Green" title="E22">A60</font></i>: 
( <span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="holder_1.html#E6:15_1_1_2"><i><font color="Green" title="E20">A58</font></i></a>, <a class="txt" href="holder_1.html#E7:15_1_1_2"><i><font color="Green" title="E21">A59</font></i></a>, <a class="ref" href="fcont_1.html#D1" title="FCONT_1:def.1">FCONT_1:def 1</a></i>;<br/><a NAME="E11:15_1_1_2"/><i><font color="Green" title="E23">A61</font></i>: 
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">ap</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">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" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:15_1_1_2_2_1"/><i><font color="Green" title="E24">A62</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><b>reconsider </b><font color="Maroon" title="c6">n</font> = <font color="Maroon" title="c5">x</font> as    <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>by </b><i><a class="txt" href="holder_1.html#E1:15_1_1_2_2_1"><i><font color="Green" title="E24">A62</font></i></a></i>;<br/><a NAME="E3:15_1_1_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10" title="RFUNCT_2:th.10">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" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="holder_1.html#E6:15_1_1_2"><i><font color="Green" title="E20">A58</font></i></a></i>;<br/><a NAME="E5:15_1_1_2_2_1"/><b>then </b><i><font color="Green" title="E25">A63</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="taylor_1.html#D4" title="TAYLOR_1:def.4">TAYLOR_1:def 4</a></i>;<br/><a NAME="E6:15_1_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>;<br/><a NAME="E7:15_1_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="limfunc1.html#T152" title="LIMFUNC1:th.152">LIMFUNC1:152</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E9:15_1_1_2_2_1"/><i><font color="Green" title="E26">A64</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> &amp; <font color="Maroon" title="c7">g</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E6:15_1_1_2"><i><font color="Green" title="E20">A58</font></i></a>, <a class="ref" href="rfunct_2.html#T21" title="RFUNCT_2:th.21">RFUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="txt" href="holder_1.html#E5:15_1_1_2_2_1"><i><font color="Green" title="E25">A63</font></i></a>, <a class="ref" href="taylor_1.html#D4" title="TAYLOR_1:def.4">TAYLOR_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="txt" href="holder_1.html#E9:15_1_1_2_2_1"><i><font color="Green" title="E26">A64</font></i></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c3">ap</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="txt" href="holder_1.html#E4:15"><i><font color="Green" title="E16">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">ap</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">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" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">ap</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T8" title="SEQ_1:th.8">SEQ_1:8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:15_1_1_2"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <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" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="holder_1.html#E1:15_1_1_2"><i><font color="Green" title="E17">A50</font></i></a>, <a class="txt" href="holder_1.html#E2:15_1_1_2"><i><font color="Green" title="E18">A51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:15_1_1_2_3"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="limfunc1.html#T152" title="LIMFUNC1:th.152">LIMFUNC1:152</a></i>;<br/>


</div><b>end;</b></div><b>then </b><span class="p1">(<span class="default"><a href="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="prepower.html#K11" title="PREPOWER:func.11">#R</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="txt" href="holder_1.html#E7:15_1_1_2"><i><font color="Green" title="E21">A59</font></i></a>, <a class="ref" href="taylor_1.html#D4" title="TAYLOR_1:def.4">TAYLOR_1:def 4</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">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="txt" href="holder_1.html#E1:15_1_1_2"><i><font color="Green" title="E17">A50</font></i></a>, <a class="txt" href="holder_1.html#E2:15_1_1_2"><i><font color="Green" title="E18">A51</font></i></a>, <a class="ref" href="power.html#D2" title="POWER:def.2">POWER:def 2</a></i>
;<br/><b>hence </b><a NAME="E14:15_1_1_2"/>
( <font color="Maroon" title="c3">ap</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">ap</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">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> )
 <b>by </b><i><a class="txt" href="holder_1.html#E10:15_1_1_2"><i><font color="Green" title="E22">A60</font></i></a>, <a class="txt" href="holder_1.html#E11:15_1_1_2"><i><font color="Green" title="E23">A61</font></i></a>, <a class="ref" href="seq_4.html#T35" title="SEQ_4:th.35">SEQ_4:35</a>, <a class="ref" href="seq_4.html#T36" title="SEQ_4:th.36">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" title="c3">ap</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">ap</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">a</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">p</font> )
 ;<br/>


</div>
