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

<b>let </b><font color="Maroon" title="c1">N</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c3">seq</font> be   <a href="toprns_1.html#NM1" title="TOPRNS_1:NM.1">Real_Sequence</a> of <font color="Maroon" title="c1">N</font>;<br/>





<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c3">seq</font> is <a href="toprns_1.html#V3" title="TOPRNS_1:attr.3">convergent</a>
 ;<br/>

<a NAME="E2:46"/><b>then </b><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c3">seq</font> is <a href="toprns_1.html#V3" title="TOPRNS_1:attr.3">convergent</a>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T43" target="_self" title="TOPRNS_1:th.43">Th43</a></i>;<br/>
<div><i><font color="Green" title="E35">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_1"/><i><font color="Green" title="E36">A4</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:46_1"/><b>then </b><i><font color="Green" title="E37">A5</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">N</font>
 <b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>;<br/><b>let </b><font color="Maroon" title="c4">q</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E3:46_1"/><i><font color="Green" title="E38">A6</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font>
 ;<br/><b>take </b><font color="Maroon" title="c5">k</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><b>let </b><font color="Maroon" title="c6">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E4:46_1"/>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 ;<br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><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">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">N</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="txt" href="toprns_1.html#E1:46_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="toprns_1.html#E2:46_1"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="toprns_1.html#D3" target="_self" title="TOPRNS_1:def.3">Def3</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">N</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="toprns_1.html#T28" target="_self" title="TOPRNS_1:th.28">Th28</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">N</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <span class="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T45" title="EUCLID:th.45">EUCLID:45</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T13" title="EUCLID:th.13">EUCLID:13</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">N</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="euclid.html#T10" title="EUCLID:th.10">EUCLID:10</a></i>
;<br/><b>hence </b><a NAME="E6:46_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><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">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="toprns_1.html#E3:46_1"><i><font color="Green" title="E38">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_2"/><i><font color="Green" title="E36">A7</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:46_2"/><b>then </b><i><font color="Green" title="E37">A8</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></i>;<br/><b>let </b><font color="Maroon" title="c4">q</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E3:46_2"/><i><font color="Green" title="E38">A9</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font>
 ;<br/><a NAME="E4:46_2"/><i><font color="Green" title="E39">A10</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="txt" href="toprns_1.html#E1:46_2"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></i>;<br/><a NAME="E5:46_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E6:46_2"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="toprns_1.html#E2:46_2"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="toprns_1.html#E3:46_2"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="real_1.html#T73" title="REAL_1:th.73">REAL_1:73</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">n1</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="E8:46_2"/><i><font color="Green" title="E40">A11</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c5">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="toprns_1.html#E1:46"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="toprns_1.html#D10" target="_self" title="TOPRNS_1:def.10">Def10</a></i>;<br/><b>take </b><font color="Maroon" title="c6">k</font> = <font color="Maroon" title="c5">n1</font>;<br/><b>let </b><font color="Maroon" title="c7">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E9:46_2"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font>
 ;<br/><a NAME="E10:46_2"/><b>then </b><i><font color="Green" title="E41">A12</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="toprns_1.html#E8:46_2"><i><font color="Green" title="E40">A11</font></i></a></i>;<br/><i><font color="Green" title="E42">A13</font></i>: <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><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">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="toprns_1.html#D3" target="_self" title="TOPRNS_1:def.3">Def3</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T14" title="EUCLID:th.14">EUCLID:14</a></i>
;<br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">q</font></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"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">q</font>

<br/>.= 
1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">q</font>
<b>by </b><i><a class="txt" href="toprns_1.html#E4:46_2"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c4">q</font>

;<br/><b>hence </b><a NAME="E13:46_2"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><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">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="toprns_1.html#E2:46_2"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="toprns_1.html#E10:46_2"><i><font color="Green" title="E41">A12</font></i></a>, <a class="txt" href="toprns_1.html#E11:46_2"><i><font color="Green" title="E42">A13</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:46"/>
 <a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="toprns_1.html#K2" title="TOPRNS_1:func.2">*</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq</font></span>)</span>
 <b>by </b><i><a class="txt" href="toprns_1.html#E2:46"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="toprns_1.html#E3:46"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="toprns_1.html#D10" target="_self" title="TOPRNS_1:def.10">Def10</a></i>;<br/>


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