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

<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>;<br/><b>let </b><font color="Maroon" title="c2">s1</font>, <font color="Maroon" title="c3">s2</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:113"/><i><font color="Green" title="E92">A1</font></i>: 
( <font color="Maroon" title="c2">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c3">s2</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> )
 <b>and </b><br/><a NAME="E2:113"/><i><font color="Green" title="E93">A2</font></i>: 
(  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 <b>and </b><br/><a NAME="E3:113"/><i><font color="Green" title="E94">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 <br/> ( <font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c3">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">s</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E5:113"/><i><font color="Green" title="E95">A4</font></i>: 
<font color="Maroon" title="c4">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39" title="SEQ_4:th.39">SEQ_4:39</a></i>;<br/>
<i><font color="Green" title="E96">A5</font></i>:  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s</font> = 
<font color="Maroon" title="c4">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
<b>by </b><i><a class="ref" href="seq_4.html#T41" title="SEQ_4:th.41">SEQ_4:41</a></i>
<br/>.= 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E7:113"/>
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="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="prepower.html#E3:113"><i><font color="Green" title="E94">A3</font></i></a></i>;<br/>
<a NAME="E8:113"/><b>then </b><i><font color="Green" title="E97">A6</font></i>: 
<font color="Maroon" title="c2">s1</font> <a href="seq_1.html#NR1" title="SEQ_1:NR.1">is_not_0</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></i>;<br/>
<a NAME="E9:113"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c4">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="prepower.html#E2:113"><i><font color="Green" title="E93">A2</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E10:113"/><b>then </b><i><font color="Green" title="E98">A7</font></i>: 
<font color="Maroon" title="c4">s</font> <a href="seq_1.html#NR1" title="SEQ_1:NR.1">is_not_0</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></i>;<br/>
<a NAME="E11:113"/><b>then </b><i><font color="Green" title="E99">A8</font></i>: 
<font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="prepower.html#E1:113"><i><font color="Green" title="E92">A1</font></i></a>, <a class="txt" href="prepower.html#E2:113"><i><font color="Green" title="E93">A2</font></i></a>, <a class="txt" href="prepower.html#E5:113"><i><font color="Green" title="E95">A4</font></i></a>, <a class="txt" href="prepower.html#E6:113"><i><font color="Green" title="E96">A5</font></i></a>, <a class="ref" href="seq_2.html#T37" title="SEQ_2:th.37">SEQ_2:37</a></i>;<br/>
<i><font color="Green" title="E100">A9</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">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></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">s1</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s</font></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:113"><i><font color="Green" title="E92">A1</font></i></a>, <a class="txt" href="prepower.html#E2:113"><i><font color="Green" title="E93">A2</font></i></a>, <a class="txt" href="prepower.html#E5:113"><i><font color="Green" title="E95">A4</font></i></a>, <a class="txt" href="prepower.html#E6:113"><i><font color="Green" title="E96">A5</font></i></a>, <a class="txt" href="prepower.html#E10:113"><i><font color="Green" title="E98">A7</font></i></a>, <a class="ref" href="seq_2.html#T38" title="SEQ_2:th.38">SEQ_2:38</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="prepower.html#E2:113"><i><font color="Green" title="E93">A2</font></i></a>, <a class="txt" href="prepower.html#E6:113"><i><font color="Green" title="E96">A5</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
;<br/>
<a NAME="E13:113"/><i><font color="Green" title="E101">A10</font></i>: 
<font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="prepower.html#E1:113"><i><font color="Green" title="E92">A1</font></i></a>, <a class="txt" href="prepower.html#E2:113"><i><font color="Green" title="E93">A2</font></i></a>, <a class="txt" href="prepower.html#E5:113"><i><font color="Green" title="E95">A4</font></i></a>, <a class="txt" href="prepower.html#E8:113"><i><font color="Green" title="E97">A6</font></i></a>, <a class="ref" href="seq_2.html#T37" title="SEQ_2:th.37">SEQ_2:37</a></i>;<br/>
<i><font color="Green" title="E102">A11</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="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></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="c4">s</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:113"><i><font color="Green" title="E92">A1</font></i></a>, <a class="txt" href="prepower.html#E2:113"><i><font color="Green" title="E93">A2</font></i></a>, <a class="txt" href="prepower.html#E5:113"><i><font color="Green" title="E95">A4</font></i></a>, <a class="txt" href="prepower.html#E8:113"><i><font color="Green" title="E97">A6</font></i></a>, <a class="ref" href="seq_2.html#T38" title="SEQ_2:th.38">SEQ_2:38</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="prepower.html#E2:113"><i><font color="Green" title="E93">A2</font></i></a>, <a class="txt" href="prepower.html#E6:113"><i><font color="Green" title="E96">A5</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
;<br/>
<b>consider </b><font color="Maroon" title="c5">m0</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="E16:113"/><i><font color="Green" title="E103">A12</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">m0</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T10" title="SEQ_4:th.10">SEQ_4:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">q</font> = <font color="Maroon" title="c5">m0</font> as   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c5">m0</font>;<br/>
<b>consider </b><font color="Maroon" title="c7">s3</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E19:113"/><i><font color="Green" title="E104">A13</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="c7">s3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c5">m0</font>;<br/>
<b>consider </b><font color="Maroon" title="c8">s4</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E21:113"/><i><font color="Green" title="E105">A14</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="c8">s4</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E22:113"/>
( <font color="Maroon" title="c7">s3</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c7">s3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c5">m0</font> )
 
<b>by </b><i><a class="txt" href="prepower.html#E13:113"><i><font color="Green" title="E101">A10</font></i></a>, <a class="txt" href="prepower.html#E14:113"><i><font color="Green" title="E102">A11</font></i></a>, <a class="txt" href="prepower.html#E19:113"><i><font color="Green" title="E104">A13</font></i></a>, <a class="ref" href="prepower.html#T26" target="_self" title="PREPOWER:th.26">Th26</a></i>;<br/>
<a NAME="E23:113"/><b>then </b><i><font color="Green" title="E106">A15</font></i>: 
( <font color="Maroon" title="c7">s3</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c7">s3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></i>;<br/>
<a NAME="E24:113"/>
( <font color="Maroon" title="c8">s4</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="c8">s4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c5">m0</font> )
 
<b>by </b><i><a class="txt" href="prepower.html#E11:113"><i><font color="Green" title="E99">A8</font></i></a>, <a class="txt" href="prepower.html#E12:113"><i><font color="Green" title="E100">A9</font></i></a>, <a class="txt" href="prepower.html#E21:113"><i><font color="Green" title="E105">A14</font></i></a>, <a class="ref" href="prepower.html#T26" target="_self" title="PREPOWER:th.26">Th26</a></i>;<br/>
<a NAME="E25:113"/><b>then </b><i><font color="Green" title="E107">A16</font></i>: 
( <font color="Maroon" title="c8">s4</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="c8">s4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></i>;<br/>
<a NAME="E26:113"/>
<font color="Maroon" title="c3">s2</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 
<b>by </b><i><a class="txt" href="prepower.html#E1:113"><i><font color="Green" title="E92">A1</font></i></a>, <a class="ref" href="seq_2.html#T27" title="SEQ_2:th.27">SEQ_2:27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">d</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E28:113"/><i><font color="Green" title="E108">A17</font></i>: 
( <font color="Maroon" title="c9">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &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  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">d</font> ) )
 <b>by </b><i><a class="ref" href="seq_2.html#T15" title="SEQ_2:th.15">SEQ_2:15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">c</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="E1:113_4"/><i><font color="Green" title="E109">A18</font></i>: 
<font color="Maroon" title="c10">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><a NAME="E2:113_4"/><b>then </b>
<font color="Maroon" title="c10">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E28:113"><i><font color="Green" title="E108">A17</font></i></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">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="E4:113_4"/><i><font color="Green" title="E110">A19</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="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">m1</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="c7">s3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">d</font>
 <b>by </b><i><a class="txt" href="prepower.html#E23:113"><i><font color="Green" title="E106">A15</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/><a NAME="E5:113_4"/><i><font color="Green" title="E111">A20</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E2:113"><i><font color="Green" title="E93">A2</font></i></a>, <a class="ref" href="prepower.html#T63" target="_self" title="PREPOWER:th.63">Th63</a></i>;<br/><a NAME="E6:113_4"/><b>then </b><i><font color="Green" title="E112">A21</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E7:113_4"/><b>then </b>
<font color="Maroon" title="c10">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E1:113_4"><i><font color="Green" title="E109">A18</font></i></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">m2</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="E9:113_4"/><i><font color="Green" title="E113">A22</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="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">m2</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="c8">s4</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E25:113"><i><font color="Green" title="E107">A16</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="c13">n</font> = <font color="Maroon" title="c11">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c12">m2</font>;<br/><b>let </b><font color="Maroon" title="c14">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="E10:113_4"/><i><font color="Green" title="E114">A23</font></i>: 
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c13">n</font>
 ;<br/><a NAME="E11:113_4"/>
<font color="Maroon" title="c11">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E12:113_4"/><b>then </b><i><font color="Green" title="E115">A24</font></i>: 
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">m1</font>
 <b>by </b><i><a class="txt" href="prepower.html#E10:113_4"><i><font color="Green" title="E114">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:113_4"/>
<font color="Maroon" title="c12">m2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E14:113_4"/><b>then </b><i><font color="Green" title="E116">A25</font></i>: 
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">m2</font>
 <b>by </b><i><a class="txt" href="prepower.html#E10:113_4"><i><font color="Green" title="E114">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:113_4_1_1"/>
( <font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font> or <font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font> <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">s1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_4_1_1_1"><b>suppose </b></a><a NAME="E1:113_4_1_1_1"/><i><font color="Green" title="E117">A26</font></i>: 
<font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font>
 ;<br/><div class="add"><a NAME="E2:113_4_1_1_1"/><i><font color="Green" title="E118">A27</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E2:113"><i><font color="Green" title="E93">A2</font></i></a>, <a class="ref" href="prepower.html#T63" target="_self" title="PREPOWER:th.63">Th63</a></i>;<br/><a NAME="E3:113_4_1_1_1"/><i><font color="Green" title="E119">A28</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E5:113_4"><i><font color="Green" title="E111">A20</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E4:113_4_1_1_1"/><i><font color="Green" title="E120">A29</font></i>: 
<font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E3:113"><i><font color="Green" title="E94">A3</font></i></a></i>;<br/><i><font color="Green" title="E121">A30</font></i>:  <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">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> = 
 <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"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E3:113"><i><font color="Green" title="E94">A3</font></i></a></i>
<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"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E2:113_4_1_1_1"><i><font color="Green" title="E118">A27</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<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"><span class="p4">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>
<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"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T121" title="XCMPLX_1:th.121">XCMPLX_1:121</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E2:113_4_1_1_1"><i><font color="Green" title="E118">A27</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E2:113"><i><font color="Green" title="E93">A2</font></i></a>, <a class="txt" href="prepower.html#E4:113_4_1_1_1"><i><font color="Green" title="E120">A29</font></i></a>, <a class="ref" href="prepower.html#T69" target="_self" title="PREPOWER:th.69">Th69</a></i>
;<br/><i><font color="Green" title="E122">A31</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D8" title="SEQ_1:def.8">SEQ_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font>
<b>by </b><i><a class="ref" href="seq_1.html#D9" title="SEQ_1:def.9">SEQ_1:def 9</a></i>
;<br/><a NAME="E7:113_4_1_1_1"/>
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E2:113"><i><font color="Green" title="E93">A2</font></i></a>, <a class="ref" href="xreal_1.html#T124" title="XREAL_1:th.124">XREAL_1:124</a></i>;<br/><a NAME="E8:113_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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">s1</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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">s1</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E1:113_4_1_1_1"><i><font color="Green" title="E117">A26</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E9:113_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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">s1</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E2:113"><i><font color="Green" title="E93">A2</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E10:113_4_1_1_1"/><b>then </b><i><font color="Green" title="E123">A32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E6:113_4_1_1_1"><i><font color="Green" title="E122">A31</font></i></a>, <a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E11:113_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c6">q</font>
 <b>by </b><i><a class="txt" href="prepower.html#E16:113"><i><font color="Green" title="E103">A12</font></i></a>, <a class="ref" href="prepower.html#T74" target="_self" title="PREPOWER:th.74">Th74</a></i>;<br/><a NAME="E12:113_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c5">m0</font>
 <b>by </b><i><a class="txt" href="prepower.html#E10:113_4_1_1_1"><i><font color="Green" title="E123">A32</font></i></a>, <a class="ref" href="prepower.html#T60" target="_self" title="PREPOWER:th.60">Th60</a></i>;<br/><a NAME="E13:113_4_1_1_1"/><b>then </b><i><font color="Green" title="E124">A33</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c5">m0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E14:113_4_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E2:113"><i><font color="Green" title="E93">A2</font></i></a>, <a class="txt" href="prepower.html#E10:113_4_1_1_1"><i><font color="Green" title="E123">A32</font></i></a>, <a class="ref" href="prepower.html#T71" target="_self" title="PREPOWER:th.71">Th71</a></i>;<br/><a NAME="E15:113_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E16:113_4_1_1_1"/><b>then </b><i><font color="Green" title="E125">A34</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E17:113_4_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c5">m0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E10:113_4_1_1_1"><i><font color="Green" title="E123">A32</font></i></a>, <a class="ref" href="prepower.html#T19" target="_self" title="PREPOWER:th.19">Th19</a></i>;<br/><a NAME="E18:113_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c5">m0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E19:113_4_1_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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c5">m0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E13:113_4_1_1_1"><i><font color="Green" title="E124">A33</font></i></a>, <a class="txt" href="prepower.html#E16:113_4_1_1_1"><i><font color="Green" title="E125">A34</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E20:113_4_1_1_1"/><b>then </b><i><font color="Green" title="E126">A35</font></i>: 
 <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"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="c8">s4</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E21:113"><i><font color="Green" title="E105">A14</font></i></a></i>;<br/><a NAME="E21:113_4_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="c8">s4</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E9:113_4"><i><font color="Green" title="E113">A22</font></i></a>, <a class="txt" href="prepower.html#E14:113_4"><i><font color="Green" title="E116">A25</font></i></a></i>;<br/><a NAME="E22:113_4_1_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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E20:113_4_1_1_1"><i><font color="Green" title="E126">A35</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E23:113_4_1_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="c3">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</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> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E6:113_4"><i><font color="Green" title="E112">A21</font></i></a>, <a class="txt" href="prepower.html#E5:113_4_1_1_1"><i><font color="Green" title="E121">A30</font></i></a>, <a class="txt" href="prepower.html#E6:113_4_1_1_1"><i><font color="Green" title="E122">A31</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E24:113_4_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">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</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">c</font>
 <b>by </b><i><a class="txt" href="prepower.html#E3:113_4_1_1_1"><i><font color="Green" title="E119">A28</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_4_1_1_2"><b>suppose </b></a><a NAME="E1:113_4_1_1_2"/><i><font color="Green" title="E117">A36</font></i>: 
<font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font> <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">s1</font>
 ;<br/><div class="add"><a NAME="E2:113_4_1_1_2"/><i><font color="Green" title="E118">A37</font></i>: 
<font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E3:113"><i><font color="Green" title="E94">A3</font></i></a></i>;<br/><a NAME="E3:113_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="prepower.html#T63" target="_self" title="PREPOWER:th.63">Th63</a></i>;<br/><a NAME="E4:113_4_1_1_2"/><b>then </b><i><font color="Green" title="E119">A38</font></i>: 
 <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">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E5:113_4_1_1_2"/><i><font color="Green" title="E120">A39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E2:113_4_1_1_2"><i><font color="Green" title="E118">A37</font></i></a>, <a class="ref" href="prepower.html#T63" target="_self" title="PREPOWER:th.63">Th63</a></i>;<br/><a NAME="E6:113_4_1_1_2"/><i><font color="Green" title="E121">A40</font></i>: 
<font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E3:113"><i><font color="Green" title="E94">A3</font></i></a></i>;<br/><i><font color="Green" title="E122">A41</font></i>:  <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">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> = 
 <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"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E3:113"><i><font color="Green" title="E94">A3</font></i></a></i>
<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"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E5:113_4_1_1_2"><i><font color="Green" title="E120">A39</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<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"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>
<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">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T121" title="XCMPLX_1:th.121">XCMPLX_1:121</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E5:113_4_1_1_2"><i><font color="Green" title="E120">A39</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E2:113"><i><font color="Green" title="E93">A2</font></i></a>, <a class="txt" href="prepower.html#E2:113_4_1_1_2"><i><font color="Green" title="E118">A37</font></i></a>, <a class="ref" href="prepower.html#T69" target="_self" title="PREPOWER:th.69">Th69</a></i>
;<br/><i><font color="Green" title="E123">A42</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">s1</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D8" title="SEQ_1:def.8">SEQ_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font>
<b>by </b><i><a class="ref" href="seq_1.html#D9" title="SEQ_1:def.9">SEQ_1:def 9</a></i>
;<br/><a NAME="E9:113_4_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E2:113_4_1_1_2"><i><font color="Green" title="E118">A37</font></i></a>, <a class="ref" href="xreal_1.html#T124" title="XREAL_1:th.124">XREAL_1:124</a></i>;<br/><a NAME="E10:113_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </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">s1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E1:113_4_1_1_2"><i><font color="Green" title="E117">A36</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E11:113_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E6:113_4_1_1_2"><i><font color="Green" title="E121">A40</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E12:113_4_1_1_2"/><b>then </b><i><font color="Green" title="E124">A43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E8:113_4_1_1_2"><i><font color="Green" title="E123">A42</font></i></a>, <a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E13:113_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c6">q</font>
 <b>by </b><i><a class="txt" href="prepower.html#E16:113"><i><font color="Green" title="E103">A12</font></i></a>, <a class="ref" href="prepower.html#T74" target="_self" title="PREPOWER:th.74">Th74</a></i>;<br/><a NAME="E14:113_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c5">m0</font>
 <b>by </b><i><a class="txt" href="prepower.html#E12:113_4_1_1_2"><i><font color="Green" title="E124">A43</font></i></a>, <a class="ref" href="prepower.html#T60" target="_self" title="PREPOWER:th.60">Th60</a></i>;<br/><a NAME="E15:113_4_1_1_2"/><b>then </b><i><font color="Green" title="E125">A44</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c5">m0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E16:113_4_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E2:113"><i><font color="Green" title="E93">A2</font></i></a>, <a class="txt" href="prepower.html#E12:113_4_1_1_2"><i><font color="Green" title="E124">A43</font></i></a>, <a class="ref" href="prepower.html#T71" target="_self" title="PREPOWER:th.71">Th71</a></i>;<br/><a NAME="E17:113_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E18:113_4_1_1_2"/><b>then </b><i><font color="Green" title="E126">A45</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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"><span class="p4">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E19:113_4_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c5">m0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E12:113_4_1_1_2"><i><font color="Green" title="E124">A43</font></i></a>, <a class="ref" href="prepower.html#T19" target="_self" title="PREPOWER:th.19">Th19</a></i>;<br/><a NAME="E20:113_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c5">m0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E21:113_4_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"><span class="p4">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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"><span class="p4">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c5">m0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E15:113_4_1_1_2"><i><font color="Green" title="E125">A44</font></i></a>, <a class="txt" href="prepower.html#E18:113_4_1_1_2"><i><font color="Green" title="E126">A45</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E22:113_4_1_1_2"/><b>then </b><i><font color="Green" title="E127">A46</font></i>: 
 <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"><span class="p4">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="c7">s3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E19:113"><i><font color="Green" title="E104">A13</font></i></a></i>;<br/><a NAME="E23:113_4_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="c7">s3</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">d</font>
 <b>by </b><i><a class="txt" href="prepower.html#E4:113_4"><i><font color="Green" title="E110">A19</font></i></a>, <a class="txt" href="prepower.html#E12:113_4"><i><font color="Green" title="E115">A24</font></i></a></i>;<br/><a NAME="E24:113_4_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"><span class="p4">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">d</font>
 <b>by </b><i><a class="txt" href="prepower.html#E22:113_4_1_1_2"><i><font color="Green" title="E127">A46</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E25:113_4_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"><font color="Maroon" title="c3">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</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> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E4:113_4_1_1_2"><i><font color="Green" title="E119">A38</font></i></a>, <a class="txt" href="prepower.html#E7:113_4_1_1_2"><i><font color="Green" title="E122">A41</font></i></a>, <a class="txt" href="prepower.html#E8:113_4_1_1_2"><i><font color="Green" title="E123">A42</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E26:113_4_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"><font color="Maroon" title="c3">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</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> <span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">d</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>;<br/><a NAME="E27:113_4_1_1_2"/><b>then </b><i><font color="Green" title="E128">A47</font></i>: 
 <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">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</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">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">d</font></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>;<br/><a NAME="E28:113_4_1_1_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">d</font>
 <b>by </b><i><a class="txt" href="prepower.html#E28:113"><i><font color="Green" title="E108">A17</font></i></a></i>;<br/><a NAME="E29:113_4_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"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">d</font>
 <b>by </b><i><a class="txt" href="prepower.html#E3:113"><i><font color="Green" title="E94">A3</font></i></a></i>;<br/><a NAME="E30:113_4_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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">d</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">d</font>
 <b>by </b><i><a class="txt" href="prepower.html#E28:113"><i><font color="Green" title="E108">A17</font></i></a>, <a class="ref" href="real_1.html#T73" title="REAL_1:th.73">REAL_1:73</a></i>;<br/><a NAME="E31:113_4_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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E28:113"><i><font color="Green" title="E108">A17</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>;<br/><a NAME="E32:113_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c9">d</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E1:113_4"><i><font color="Green" title="E109">A18</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E33:113_4_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">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</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">c</font>
 <b>by </b><i><a class="txt" href="prepower.html#E27:113_4_1_1_2"><i><font color="Green" title="E128">A47</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E16:113_4"/>
 <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">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</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">c</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E30:113"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">s1</font></span>)</span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="prepower.html#E1:113"><i><font color="Green" title="E92">A1</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/>


</div>
