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<b>let </b><font color="Maroon" title="c1">seq</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> implies  <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a>  )</span><br/>

<b>assume </b><a NAME="E1:6"/>
<font color="Maroon" title="c1">seq</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span><br/>

<a NAME="E2:6"/><b>then </b><i><font color="Green" title="E7">A1</font></i>: 
 <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c1">seq</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D10" title="COMSEQ_3:def.10">COMSEQ_3:def 10</a></i>;<br/>
 <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span> = 
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D8" title="COMSEQ_3:def.8">COMSEQ_3:def 8</a></i>
<br/>.= 
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span>
<b>by </b><i><a class="txt" href="csspace2.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c1">seq</font></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<b>by </b><i><a class="txt" href="csspace2.html#E2:6"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="comseq_2.html#T12" title="COMSEQ_2:th.12">COMSEQ_2:12</a></i>
;<br/>
<b>hence </b><a NAME="E4:6"/>
 <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_2.html#K1" title="COMSEQ_2:func.1">*'</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
 <b>by </b><i><a class="ref" href="comseq_3.html#D8" title="COMSEQ_3:def.8">COMSEQ_3:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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