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

<b>let </b><font color="Maroon">s</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><b>let </b><font color="Maroon">s'</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">s</font> is <a href="comseq_2.html#V2" target="_self">convergent</a> &amp; <font color="Maroon">s'</font> is <a href="comseq_2.html#V2" target="_self">convergent</a> )
 ;<br/>

<a NAME="E2:24"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">s</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">s'</font> is <a href="comseq_2.html#V2" target="_self">convergent</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:24"/>
for <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</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">s</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">s'</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:24_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:24_1"/><b>then </b><i><font color="Green">E38</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<b>then consider </b><font color="Maroon">n1</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="E4:24_1"/><i><font color="Green">E40</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="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">n2</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="E6:24_1"/><i><font color="Green">E41</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="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s'</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> =  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T1">FINSEQ_2:1</a></i>;<br/>
<b>take </b>
<font color="Maroon">n</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="E8:24_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E9:24_1"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E10:24_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E11:24_1"/><b>then </b>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E12:24_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i/>;<br/>
<a NAME="E13:24_1"/>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E14:24_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E15:24_1"/><b>then </b>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E16:24_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s'</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i/>;<br/>
<a NAME="E17:24_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s'</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</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">p</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<span class="p1"><a href="complex1.html#K17">|.</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">s</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">s'</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17">|.</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">s</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">s'</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D4">COMSEQ_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</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">s</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s'</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>

;<br/>
<a NAME="E19:24_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</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">s</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">s'</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s'</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="complex1.html#T142">COMPLEX1:142</a></i>;<br/>
<a NAME="E20:24_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s'</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="complex1.html#K17">|.</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">s</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">s'</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s'</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E21:24_1"/>
<span class="p1"><a href="complex1.html#K17">|.</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">s</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">s'</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s'</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:24"/>
 <a href="comseq_2.html#K2" target="_self">lim</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">s'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">s'</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div>
