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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="clopban3.html#V1" target="_self">summable</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="clopban3.html#V1" target="_self">summable</a> )
 ;<br/>

<a NAME="E2:35"/><i><font color="Green">E31</font></i>: 
(  <a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font> is <a href="clvect_1.html#V6">convergent</a> &amp;  <a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font> is <a href="clvect_1.html#V6">convergent</a> )
 
<b>by </b><i><a class="txt" href="clopban3.html#E1:35"><i><font color="Green">E30</font></i></a>, <a class="ref" href="clopban3.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E3:35"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="clvect_1.html#K7">-</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="clvect_1.html#V6">convergent</a>
 
<b>by </b><i><a class="ref" href="clvect_1.html#T116">CLVECT_1:116</a></i>;<br/>
<a NAME="E4:35"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="clvect_1.html#K7">-</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="clopban3.html#T20" target="_self">Th20</a></i>;<br/>
 <a href="clopban3.html#K2" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
 <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>

<br/>.= 
 <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">c<sub>2</sub></font></span>)</span> <a href="clvect_1.html#K7">-</a> <span class="p2">(<span class="default"><a href="clopban3.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clopban3.html#T20" target="_self">Th20</a></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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</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">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="clopban3.html#E2:35"><i><font color="Green">E31</font></i></a>, <a class="ref" href="clvect_1.html#T122">CLVECT_1:122</a></i>
;<br/>
<b>then </b> <a href="clopban3.html#K2" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="clopban3.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</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">c<sub>3</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="clopban3.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="clopban3.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

;<br/>
<b>hence </b><a NAME="E7:35"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="clopban3.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="clopban3.html#E3:35"><i><font color="Green">E32</font></i></a>, <a class="txt" href="clopban3.html#E4:35"><i><font color="Green">E33</font></i></a>, <a class="ref" href="clopban3.html#D2" target="_self">Def2</a></i>;<br/>


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