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

<b>let </b><font color="Maroon" title="c1">c</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">seq</font>, <font color="Maroon" title="c3">seq1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c2">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c3">seq1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">rseq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E3:16"/><i><font color="Green" title="E16">A3</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">cseq</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c1">c</font> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E5:16"/><i><font color="Green" title="E17">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c5">cseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">c</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">i</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>thus </b><font color="Maroon" title="c4">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="rsspace3.html#E3:16"><i><font color="Green" title="E16">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">cseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="rsspace3.html#E5:16"><i><font color="Green" title="E17">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">cseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_1.html#K17" title="SEQ_1:func.17">-</a> <font color="Maroon" title="c5">cseq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T14" title="SEQ_1:th.14">SEQ_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p4">(<span class="default"><a href="seq_1.html#K17" title="SEQ_1:func.17">-</a> <font color="Maroon" title="c5">cseq</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c5">cseq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T15" title="SEQ_1:th.15">SEQ_1:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22" title="SEQ_1:func.22">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c5">cseq</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T16" title="SEQ_1:th.16">SEQ_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22" title="SEQ_1:func.22">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c5">cseq</font></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E7:16"/><b>then </b>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c4">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22" title="SEQ_1:func.22">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c5">cseq</font></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font>
 
;<br/>
<a NAME="E8:16"/><b>then </b><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c4">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22" title="SEQ_1:func.22">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c5">cseq</font></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c3">seq1</font>
 
<b>by </b><i><a class="ref" href="seq_1.html#T8" title="SEQ_1:th.8">SEQ_1:8</a></i>;<br/>
<a NAME="E9:16"/><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon" title="c5">cseq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39" title="SEQ_4:th.39">SEQ_4:39</a></i>;<br/>
<a NAME="E10:16"/><b>then </b><i><font color="Green" title="E20">A7</font></i>: 
<font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c5">cseq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="rsspace3.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="seq_2.html#T25" title="SEQ_2:th.25">SEQ_2:25</a></i>;<br/>
<a NAME="E11:16"/><b>then </b><i><font color="Green" title="E21">A8</font></i>: 
 <a href="seq_1.html#K22" title="SEQ_1:func.22">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c5">cseq</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T26" title="SEQ_4:th.26">SEQ_4:26</a></i>;<br/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22" title="SEQ_1:func.22">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c5">cseq</font></span>)</span></span>)</span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c5">cseq</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rsspace3.html#E10:16"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="seq_4.html#T27" title="SEQ_4:th.27">SEQ_4:27</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c5">cseq</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rsspace3.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="rsspace3.html#E9:16"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="seq_2.html#T26" title="SEQ_2:th.26">SEQ_2:26</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">cseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_4.html#T41" title="SEQ_4:th.41">SEQ_4:41</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">c</font></span>)</span>
<b>by </b><i><a class="txt" href="rsspace3.html#E5:16"><i><font color="Green" title="E17">A4</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E13:16"/>
( <font color="Maroon" title="c4">rseq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c4">rseq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">c</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">seq1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rsspace3.html#E2:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="rsspace3.html#E8:16"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="rsspace3.html#E11:16"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="seq_2.html#T19" title="SEQ_2:th.19">SEQ_2:19</a>, <a class="ref" href="seq_2.html#T20" title="SEQ_2:th.20">SEQ_2:20</a></i>;<br/>


</div>
