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<span class="kw">let </span><font color="Maroon" title="c3">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="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c3">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b1">n</font> is  <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a>  implies <font color="Maroon" title="c3">seq</font> is  <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a>  )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c4">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><a NAME="E2:83"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<font color="Maroon" title="c3">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c4">n</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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">seq</font> is  <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> </span><br/>

<a NAME="E3:83"/>
 <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#T23" target="_self" title="COMSEQ_3:th.23">Th23</a></span>;<br/>
<a NAME="E4:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A2</font></span>: 
 <a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <font color="Maroon" title="c3">seq</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="series_1.html#T16" title="SERIES_1:th.16">SERIES_1:16</a></span>;<br/>
<a NAME="E5:83"/>
 <a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#T23" target="_self" title="COMSEQ_3:th.23">Th23</a></span>;<br/>
<a NAME="E6:83"/><span class="kw">then </span>
 <a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <font color="Maroon" title="c3">seq</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="series_1.html#T16" title="SERIES_1:th.16">SERIES_1:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:83"/>
<font color="Maroon" title="c3">seq</font> is  <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="comseq_3.html#T58" target="_self" title="COMSEQ_3:th.58">Th58</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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