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<b>let </b><font color="Maroon">seq</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:6"/>
<font color="Maroon">seq</font> is <a href="comseq_3.html#V1">summable</a>
 ;<br/>

<a NAME="E2:6"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font> is <a href="comseq_2.html#V2">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D10">COMSEQ_3:def 10</a></i>;<br/>
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_2.html#K1">*'</a> </span>)</span> = 
 <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D8">COMSEQ_3:def 8</a></i>
<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">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span>
<b>by </b><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">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i>, <a class="ref" href="comseq_2.html#T12">COMSEQ_2:12</a></i>
;<br/>
<b>hence </b><a NAME="E4:6"/>
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <font color="Maroon">seq</font></span>)</span> <a href="complex1.html#K15">*'</a> 
 <b>by </b><i><a class="ref" href="comseq_3.html#D8">COMSEQ_3:def 8</a></i>;<br/>


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