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

<b>let </b><font color="Maroon">CNS</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">S1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">CNS</font>;<br/><b>let </b><font color="Maroon">S2</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">CNS</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:149"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">S1</font> is <a href="clvect_1.html#V6" target="_self">convergent</a>
 <b>and </b><br/><a NAME="E2:149"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S2</font> is <a href="clvect_1.html#V6" target="_self">convergent</a>
 ;<br/>

<a NAME="E3:149"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S1</font> <a href="clvect_1.html#K6" target="_self">+</a> <font color="Maroon">S2</font> is <a href="clvect_1.html#V6" target="_self">convergent</a>
 
<b>by </b><i><a class="ref" href="clvect_1.html#T24" target="_self">Th24</a>, <a class="ref" href="clvect_1.html#D4" target="_self">Def4</a>, </i>;<br/>
<b>set </b><font color="Maroon">g1</font> =  <a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S1</font>;<br/>
<b>set </b><font color="Maroon">g2</font> =  <a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S2</font>;<br/>
<b>set </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S2</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="149_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:149_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E2:149_1"/><b>then </b><i><font color="Green">E37</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="seq_2.html#T3">SEQ_2:3</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="E4:149_1"/><i><font color="Green">E42</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>   st <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1"><a href="clvect_1.html#K5" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S1</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="clvect_1.html#T24" target="_self">Th24</a>, <a class="ref" href="clvect_1.html#T22" target="_self">Th22</a></i>;<br/><b>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="E6:149_1"/><i><font color="Green">E43</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>   st <font color="Maroon">m2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1"><a href="clvect_1.html#K5" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S2</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="clvect_1.html#D4" target="_self">Def4</a>, , <a class="ref" href="clvect_1.html#T22" target="_self">Th22</a></i>;<br/><b>take </b><font color="Maroon">k</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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E7:149_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E8:149_1"/>
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E9:149_1"/><b>then </b>
<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="clvect_1.html#T29" target="_self">Th29</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:149_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S1</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i/>;<br/><a NAME="E11:149_1"/>
<font color="Maroon">m2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E12:149_1"/><b>then </b>
<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="clvect_1.html#T29" target="_self">Th29</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:149_1"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S2</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i/>;<br/><a NAME="E14:149_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S1</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S2</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="clvect_1.html#T30" target="_self">Th30</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><i><font color="Green">E76</font></i>: <span class="p1"><a href="clvect_1.html#K5" target="_self">||.</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">S1</font> <a href="clvect_1.html#K6" target="_self">+</a> <font color="Maroon">S2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S2</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span> = 
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p5">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S2</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S2</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T45">RLVECT_1:45</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</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">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p0">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p0">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</a><span class="default"><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="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S2</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</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">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S2</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/><a NAME="E16:149_1"/>
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</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">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S2</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S1</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S2</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span>
 <b>by </b><i><a class="txt" href="clvect_1.html#E10:149_1"><i><font color="Green">E74</font></i></a></i>;<br/><b>hence </b><a NAME="E17:149_1"/>
<span class="p1"><a href="clvect_1.html#K5" target="_self">||.</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">S1</font> <a href="clvect_1.html#K6" target="_self">+</a> <font color="Maroon">S2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S2</font></span>)</span></span>)</span></span>)</span></span><a href="clvect_1.html#K5" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="clvect_1.html#T31" target="_self">Th31</a>, <a class="ref" href="clvect_1.html#T33" target="_self">Th33</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:149"/>
 <a href="clvect_1.html#K11" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="clvect_1.html#K6" target="_self">+</a> <font color="Maroon">S2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11" target="_self">lim</a> <font color="Maroon">S2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="clvect_1.html#T22" target="_self">Th22</a></i>;<br/>


</div>
