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<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:45"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="comseq_2.html#V2" target="_self">convergent</a>
 <b>and </b><br/><a NAME="E2:45"/><i><font color="Green">E26</font></i>: 
 <a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 ;<br/>

<a NAME="E3:45"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="txt" href="comseq_2.html#E2:45"><i><font color="Green">E26</font></i></a>, <a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/>
<a NAME="E4:45"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T217">XREAL_1:217</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:45"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></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"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="comseq_2.html#E1:45"><i><font color="Green">E25</font></i></a>, <a class="ref" href="comseq_2.html#D5" target="_self">Def5</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E7:45"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E8:45"/><b>then </b><i><font color="Green">E28</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">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></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"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="comseq_2.html#E6:45"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E9:45"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></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"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="complex1.html#T145">COMPLEX1:145</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"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></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"><a href="complex1.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>

<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">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
;<br/>
<a NAME="E11:45"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="comseq_2.html#E8:45"><i><font color="Green">E28</font></i></a>, <a class="txt" href="comseq_2.html#E9:45"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:45"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> 2</span>)</span> <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"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
;<br/>
<a NAME="E13:45"/>
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span> <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"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> 2
 
;<br/>
<b>hence </b><a NAME="E14:45"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2" target="_self">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="txt" href="comseq_2.html#E11:45"><i><font color="Green">E30</font></i></a>, <a class="txt" href="comseq_2.html#E12:45"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>


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