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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K10" title="COMSEQ_3:func.10">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#K10" title="COMSEQ_3:func.10">Sum</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:6"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="comseq_3.html#K10" title="COMSEQ_3:func.10">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#K10" title="COMSEQ_3:func.10">Sum</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span><br/>

 <a href="comseq_3.html#K10" title="COMSEQ_3:func.10">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="series_1.html#K2" title="SERIES_1:func.2">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D7" title="COMSEQ_3:def.7">COMSEQ_3:def 7</a></span>
<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="series_1.html#K2" title="SERIES_1:func.2">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="csspace2.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>
<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="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <font color="Maroon" title="c1">seq</font></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="comseq_2.html#T12" title="COMSEQ_2:th.12">COMSEQ_2:12</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:6"/>
 <a href="comseq_3.html#K10" title="COMSEQ_3:func.10">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#K10" title="COMSEQ_3:func.10">Sum</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D7" title="COMSEQ_3:def.7">COMSEQ_3:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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