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

<b>let </b><font color="Maroon">p</font> be   <a href="rat_1.html#NM1">Rational</a>;<br/><b>let </b><font color="Maroon">s1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">s2</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:115"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">s1</font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">s2</font> is <a href="seq_2.html#V4">convergent</a> )
 <b>and </b><br/><a NAME="E2:115"/><i><font color="Green">E32</font></i>: 
(  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>and </b><br/><a NAME="E3:115"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">s</font> = <a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E5:115"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E6:115"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">s</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39">SEQ_4:39</a></i>;<br/>
<i><font color="Green">E43</font></i>:  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s</font> = 
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="seq_4.html#T41">SEQ_4:41</a></i>
<br/>.= 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font>
<b>by </b><i/>
;<br/>
<a NAME="E8:115"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E9:115"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">s1</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/>
<a NAME="E10:115"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:115"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">s</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 
<b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/>
<a NAME="E12:115"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, , , , <a class="ref" href="seq_2.html#T37">SEQ_2:37</a></i>;<br/>
<i><font color="Green">E60</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i>, , , , , <a class="ref" href="seq_2.html#T38">SEQ_2:38</a></i>
<br/>.= 
1
<b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
;<br/>
<a NAME="E14:115"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, , , , <a class="ref" href="seq_2.html#T37">SEQ_2:37</a></i>;<br/>
<i><font color="Green">E62</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="seq_2.html#T38">SEQ_2:38</a></i>
<br/>.= 
1
<b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
;<br/>
<b>consider </b><font color="Maroon">m0</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E17:115"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m0</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T10">SEQ_4:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">m0</font> as   <a href="rat_1.html#NM1">Rational</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>  <a href="real_1.html#NM1">Real</a> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">m0</font>;<br/>
<b>consider </b><font color="Maroon">s3</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E20:115"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>  <a href="real_1.html#NM1">Real</a> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">m0</font>;<br/>
<b>consider </b><font color="Maroon">s4</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E22:115"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">s4</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E23:115"/>
( <font color="Maroon">s3</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s3</font> <a href="hidden.html#R1">=</a> 1 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">m0</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E24:115"/><b>then </b><i><font color="Green">E73</font></i>: 
( <font color="Maroon">s3</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s3</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="newton.html#T15">NEWTON:15</a></i>;<br/>
<a NAME="E25:115"/>
( <font color="Maroon">s4</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s4</font> <a href="hidden.html#R1">=</a> 1 <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">m0</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E26:115"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">s4</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s4</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="newton.html#T15">NEWTON:15</a></i>;<br/>
<a NAME="E27:115"/>
<font color="Maroon">s2</font> is <a href="seq_2.html#V3">bounded</a>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T27">SEQ_2:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">d</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E29:115"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> ) )
 <b>by </b><i><a class="ref" href="seq_2.html#T15">SEQ_2:15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:115_4"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E2:115_4"/><b>then </b>
<font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="prepower.html#T29" target="_self">Th29</a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><b>then consider </b><font color="Maroon">m1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:115_4"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="prepower.html#E32"><i><font color="Green">Lemma108</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/><a NAME="E5:115_4"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E6:115_4"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E7:115_4"/><b>then </b>
<font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="prepower.html#T30" target="_self">Th30</a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><b>then consider </b><font color="Maroon">m2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:115_4"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m2</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s4</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T28" target="_self">Th28</a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m2</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E10:115_4"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E11:115_4"/>
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E12:115_4"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="prepower.html#T36" target="_self">Th36</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:115_4"/>
<font color="Maroon">m2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E14:115_4"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m2</font>
 <b>by </b><i><a class="ref" href="prepower.html#T36" target="_self">Th36</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:115_4_1_1"/>
( <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> or <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_4_1_1_1"><b>suppose </b></a><a NAME="E1:115_4_1_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font>
 ;<br/><div class="add"><a NAME="E2:115_4_1_1_1"/><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E17:115_4_1_1_1"><i><font color="Green">E94</font></i></a>, </i>;<br/><a NAME="E3:115_4_1_1_1"/><i><font color="Green">E87</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="prepower.html#T32" target="_self">Th32</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E4:115_4_1_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E21:115_4_1_1_1"><i><font color="Green">E95</font></i></a></i>;<br/><i><font color="Green">E89</font></i>:  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E21:115_4_1_1_1"><i><font color="Green">E95</font></i></a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</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">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T40" target="_self">Th40</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</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">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">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">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">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">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</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">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T121">XCMPLX_1:121</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">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">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#T40" target="_self">Th40</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">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">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E17:115_4_1_1_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="prepower.html#D4" target="_self">Def4</a>, <a class="ref" href="prepower.html#T25" target="_self">Th25</a></i>
;<br/><i><font color="Green">E90</font></i>: <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="seq_1.html#D9">SEQ_1:def 9</a></i>
;<br/><a NAME="E7:115_4_1_1_1"/>
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="real_1.html#K2">"</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="prepower.html#E17:115_4_1_1_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><a NAME="E8:115_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T39" target="_self">Th39</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E9:115_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E17:115_4_1_1_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E10:115_4_1_1_1"/><b>then </b><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="prepower.html#T45" target="_self">Th45</a>, <a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E11:115_4_1_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E12:115_4_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="prepower.html#T46" target="_self">Th46</a>, <a class="txt" href="prepower.html#E17:115"><i><font color="Green">E63</font></i></a></i>;<br/><a NAME="E13:115_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">m0</font>
 <b>by </b><i><a class="txt" href="prepower.html#E55"><i><font color="Green">Lemma126</font></i></a>, <a class="txt" href="prepower.html#E20"><i><font color="Green">Lemma49</font></i></a></i>;<br/><a NAME="E14:115_4_1_1_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">m0</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E15:115_4_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E17:115_4_1_1_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="prepower.html#T46" target="_self">Th46</a>, <a class="txt" href="prepower.html#E13:115"><i><font color="Green">E60</font></i></a></i>;<br/><a NAME="E16:115_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">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E17:115_4_1_1_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E18:115_4_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">m0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="prepower.html#T46" target="_self">Th46</a>, </i>;<br/><a NAME="E19:115_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">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">m0</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E20:115_4_1_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">m0</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T47" target="_self">Th47</a>, <a class="ref" href="prepower.html#T49" target="_self">Th49</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E21:115_4_1_1_1"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s4</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E22:115_4_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s4</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T35" target="_self">Th35</a>, <a class="ref" href="prepower.html#T38" target="_self">Th38</a></i>;<br/><a NAME="E23:115_4_1_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T50" target="_self">Th50</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E24:115_4_1_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T34" target="_self">Th34</a>, <a class="ref" href="prepower.html#T44" target="_self">Th44</a>, <a class="ref" href="prepower.html#T45" target="_self">Th45</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E25:115_4_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="prepower.html#E48"><i><font color="Green">Lemma121</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_4_1_1_2"><b>suppose </b></a><a NAME="E1:115_4_1_1_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font>
 ;<br/><div class="add"><a NAME="E2:115_4_1_1_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:115_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E4:115_4_1_1_2"/><b>then </b><i><font color="Green">E98</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E5:115_4_1_1_2"/><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="prepower.html#T40" target="_self">Th40</a>, </i>;<br/><a NAME="E6:115_4_1_1_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><i><font color="Green">E101</font></i>:  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D4" target="_self">Def4</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</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">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">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">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">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">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</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">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T121">XCMPLX_1:121</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D4" target="_self">Def4</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</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">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">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">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">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">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="prepower.html#T40" target="_self">Th40</a>, <a class="ref" href="prepower.html#T25" target="_self">Th25</a></i>
;<br/><i><font color="Green">E102</font></i>: <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="txt" href="prepower.html#E1:115_4_1_1_2"><i><font color="Green">E96</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="seq_1.html#D9">SEQ_1:def 9</a></i>
;<br/><a NAME="E9:115_4_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K2">"</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="prepower.html#T40" target="_self">Th40</a>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><a NAME="E10:115_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T39" target="_self">Th39</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E11:115_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="prepower.html#T44" target="_self">Th44</a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E12:115_4_1_1_2"/><b>then </b><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="prepower.html#T46" target="_self">Th46</a>, <a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/><a NAME="E13:115_4_1_1_2"/><b>then </b><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E14:115_4_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="prepower.html#E16:115_4_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="prepower.html#E55"><i><font color="Green">Lemma126</font></i></a>, <a class="txt" href="prepower.html#E17:115"><i><font color="Green">E63</font></i></a></i>;<br/><a NAME="E15:115_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">m0</font>
 <b>by </b><i><a class="ref" href="prepower.html#T47" target="_self">Th47</a>, <a class="txt" href="prepower.html#E20"><i><font color="Green">Lemma49</font></i></a></i>;<br/><a NAME="E16:115_4_1_1_2"/><b>then </b><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">m0</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E17:115_4_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="txt" href="prepower.html#E55"><i><font color="Green">Lemma126</font></i></a>, <a class="txt" href="prepower.html#E13:115"><i><font color="Green">E60</font></i></a></i>;<br/><a NAME="E18:115_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">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E19:115_4_1_1_2"/><b>then </b><i><font color="Green">E106</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E20:115_4_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">m0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="prepower.html#E55"><i><font color="Green">Lemma126</font></i></a>, </i>;<br/><a NAME="E21:115_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">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">m0</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E22:115_4_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K3" target="_self">|^</a> <font color="Maroon">m0</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T49" target="_self">Th49</a>, <a class="ref" href="prepower.html#T50" target="_self">Th50</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E23:115_4_1_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E19:115_4_1_1_2"><i><font color="Green">E106</font></i></a></i>;<br/><a NAME="E24:115_4_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="prepower.html#T31" target="_self">Th31</a>, <a class="ref" href="prepower.html#T37" target="_self">Th37</a></i>;<br/><a NAME="E25:115_4_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">s1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="prepower.html#T51" target="_self">Th51</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E26:115_4_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="prepower.html#E48"><i><font color="Green">Lemma121</font></i></a>, <a class="ref" href="prepower.html#T45" target="_self">Th45</a>, <a class="ref" href="prepower.html#T46" target="_self">Th46</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E27:115_4_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/><a NAME="E28:115_4_1_1_2"/><b>then </b><i><font color="Green">E194</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">d</font></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/><a NAME="E29:115_4_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="prepower.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E30:115_4_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/><a NAME="E31:115_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="prepower.html#T29" target="_self">Th29</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><a NAME="E32:115_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i><a class="ref" href="prepower.html#T29" target="_self">Th29</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><a NAME="E33:115_4_1_1_2"/><b>then </b>
<font color="Maroon">c</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">d</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c</font> <a href="xcmplx_0.html#K3">*</a> 1
 <b>by </b><i><a class="ref" href="prepower.html#T30" target="_self">Th30</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E34:115_4_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="prepower.html#T52" target="_self">Th52</a>, <a class="ref" href="xxreal_0.html#T2">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="E16:115_4"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E31:115"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">s1</font></span>)</span> <a href="prepower.html#K9" target="_self">#Q</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>


</div>
