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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">seq1</font>, <font color="Maroon" title="c3">seq2</font> be   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:34"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c2">seq1</font> is <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c3">seq2</font> is <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>
 ;<br/>

<a NAME="E3:34"/><i><font color="Green" title="E31">A3</font></i>: 
(  <a href="clopban3.html#K1" title="CLOPBAN3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq1</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> &amp;  <a href="clopban3.html#K1" title="CLOPBAN3:func.1">Partial_Sums</a> <font color="Maroon" title="c3">seq2</font> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a> )
 
<b>by </b><i><a class="txt" href="clopban3.html#E1:34"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="clopban3.html#E2:34"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="clopban3.html#D2" target="_self" title="CLOPBAN3:def.2">Def2</a></i>;<br/>
<a NAME="E4:34"/><b>then </b><i><font color="Green" title="E32">A4</font></i>: 
<span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">+</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Partial_Sums</a> <font color="Maroon" title="c3">seq2</font></span>)</span> is <a href="clvect_1.html#V6" title="CLVECT_1:attr.6">convergent</a>
 
<b>by </b><i><a class="ref" href="clvect_1.html#T115" title="CLVECT_1:th.115">CLVECT_1:115</a></i>;<br/>
<a NAME="E5:34"/><i><font color="Green" title="E33">A5</font></i>: 
<span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">+</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Partial_Sums</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="clopban3.html#K1" title="CLOPBAN3:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="clopban3.html#T19" target="_self" title="CLOPBAN3:th.19">Th19</a></i>;<br/>
 <a href="clopban3.html#K2" title="CLOPBAN3:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span> = 
 <a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">+</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Partial_Sums</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T19" target="_self" title="CLOPBAN3:th.19">Th19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11" title="CLVECT_1:func.11">lim</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Partial_Sums</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="clopban3.html#E3:34"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="clvect_1.html#T121" title="CLVECT_1:th.121">CLVECT_1:121</a></i>
;<br/>
<a NAME="E7:34"/><b>then </b>
 <a href="clopban3.html#K2" title="CLOPBAN3:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban3.html#K2" title="CLOPBAN3:func.2">Sum</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="clopban3.html#K2" title="CLOPBAN3:func.2">Sum</a> <font color="Maroon" title="c3">seq2</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E8:34"/>
( <font color="Maroon" title="c2">seq1</font> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">+</a> <font color="Maroon" title="c3">seq2</font> is <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a> &amp;  <a href="clopban3.html#K2" title="CLOPBAN3:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="clvect_1.html#K6" title="CLVECT_1:func.6">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban3.html#K2" title="CLOPBAN3:func.2">Sum</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="clopban3.html#K2" title="CLOPBAN3:func.2">Sum</a> <font color="Maroon" title="c3">seq2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="clopban3.html#E4:34"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="clopban3.html#E5:34"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="clopban3.html#D2" target="_self" title="CLOPBAN3:def.2">Def2</a></i>;<br/>


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