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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><a NAME="E1:55"/><i><font color="Green">E36</font></i>: 
(  <a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V4">convergent</a> )
 ;<br/>

<a NAME="E2:55"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<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>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E3:55"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K4">Im</a> <span class="p1">(<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>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E4:55"/><b>then </b>
 <a href="comseq_2.html#K2">lim</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="comseq_3.html#E1:55"><i><font color="Green">E36</font></i></a>, <a class="txt" href="comseq_3.html#E2:55"><i><font color="Green">E37</font></i></a>, <a class="ref" href="comseq_3.html#T39" target="_self">Th39</a></i>;<br/>
<b>hence </b><a NAME="E5:55"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="comseq_2.html#V2">convergent</a> &amp;  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a>, <a class="txt" href="comseq_3.html#E3:55"><i><font color="Green">E38</font></i></a>, <a class="txt" href="comseq_3.html#E1:55"><i><font color="Green">E36</font></i></a>, <a class="txt" href="comseq_3.html#E2:55"><i><font color="Green">E37</font></i></a>, <a class="ref" href="comseq_3.html#T39" target="_self">Th39</a></i>;<br/>


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