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<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:77"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="comseq_3.html#V1" target="_self">summable</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="comseq_3.html#V1" target="_self">summable</a> )
 ;<br/>

<a NAME="E2:77"/><b>then </b><i><font color="Green">E48</font></i>: 
(  <a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font> is <a href="comseq_2.html#V2">convergent</a> &amp;  <a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>4</sub></font> is <a href="comseq_2.html#V2">convergent</a> )
 
<b>by </b><i><a class="ref" href="comseq_3.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E3:77"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="comseq_2.html#V2">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_2.html#T13">COMSEQ_2:13</a></i>;<br/>
<a NAME="E4:77"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T27" target="_self">Th27</a></i>;<br/>
 <a href="comseq_3.html#K8" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
 <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>

<br/>.= 
 <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T27" target="_self">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="comseq_3.html#E2:77"><i><font color="Green">E48</font></i></a>, <a class="ref" href="comseq_2.html#T14">COMSEQ_2:14</a></i>
;<br/>
<b>then </b> <a href="comseq_3.html#K8" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="comseq_3.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

;<br/>
<b>hence </b><a NAME="E7:77"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> is <a href="comseq_3.html#V1" target="_self">summable</a> &amp;  <a href="comseq_3.html#K8" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="comseq_3.html#E3:77"><i><font color="Green">E49</font></i></a>, <a class="txt" href="comseq_3.html#E4:77"><i><font color="Green">E50</font></i></a>, <a class="ref" href="comseq_3.html#D10" target="_self">Def10</a></i>;<br/>


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