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<b>let </b><font color="Maroon">X</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">seq1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>, <font color="Maroon">seq2</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:34"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">seq1</font> is <a href="clopban3.html#V1" target="_self">summable</a>
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">seq2</font> is <a href="clopban3.html#V1" target="_self">summable</a>
 ;<br/>

<a NAME="E3:34"/><i><font color="Green">E35</font></i>: 
(  <a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq1</font> is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq2</font> is <a href="clvect_1.html#V6">convergent</a> )
 
<b>by </b><i><a class="ref" href="clopban3.html#D1" target="_self">Def1</a>, , </i>;<br/>
<a NAME="E4:34"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq1</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq2</font></span>)</span> is <a href="clvect_1.html#V6">convergent</a>
 
<b>by </b><i><a class="ref" href="clvect_1.html#T115">CLVECT_1:115</a></i>;<br/>
<a NAME="E5:34"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq1</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq2</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="clopban3.html#K1" target="_self">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="clvect_1.html#K6">+</a> <font color="Maroon">seq2</font></span>)</span>
 
<b>by </b><i/>;<br/>
 <a href="clopban3.html#K2" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="clvect_1.html#K6">+</a> <font color="Maroon">seq2</font></span>)</span> = 
 <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq1</font></span>)</span> <a href="clvect_1.html#K6">+</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">seq2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="clvect_1.html#T121">CLVECT_1:121</a></i>
;<br/>
<a NAME="E7:34"/><b>then </b>
 <a href="clopban3.html#K2" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="clvect_1.html#K6">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clopban3.html#K2" target="_self">Sum</a> <font color="Maroon">seq1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="clopban3.html#K2" target="_self">Sum</a> <font color="Maroon">seq2</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E8:34"/>
( <font color="Maroon">seq1</font> <a href="clvect_1.html#K6">+</a> <font color="Maroon">seq2</font> is <a href="clopban3.html#V1" target="_self">summable</a> &amp;  <a href="clopban3.html#K2" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="clvect_1.html#K6">+</a> <font color="Maroon">seq2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="clopban3.html#K2" target="_self">Sum</a> <font color="Maroon">seq1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="clopban3.html#K2" target="_self">Sum</a> <font color="Maroon">seq2</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="clopban3.html#E1:34"><i><font color="Green">E33</font></i></a>, </i>;<br/>


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