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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_3.html#K11" title="COMSEQ_3:func.11">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K7" title="RVSUM_4:func.7">Sequel</a> <font color="Maroon" title="c1">f</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c2">g</font> =  <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Re</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K7" title="RVSUM_4:func.7">Sequel</a> <font color="Maroon" title="c1">f</font></span>)</span> as   <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">n</font> =  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">f</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<span class="lab"><font color="Green" title="E47">A2</font></span>:  <a href="comseq_3.html#K11" title="COMSEQ_3:func.11">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K7" title="RVSUM_4:func.7">Sequel</a> <font color="Maroon" title="c1">f</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Im</a> <span class="p4">(<span class="default"><a href="rvsum_4.html#K7" title="RVSUM_4:func.7">Sequel</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#T53" onmouseover="rs('comseq_3/T53')" onmouseout="rh()">COMSEQ_3:53</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span>

<br/>.= 
 <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Maroon" title="c2">g</font>

;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="121_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:121_2"/>
( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="121_2_1"><span class="kw">suppose </span></a><a NAME="E1:121_2_1"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_3.html#K11" title="COMSEQ_3:func.11">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K7" title="RVSUM_4:func.7">Sequel</a> <font color="Maroon" title="c1">f</font></span>)</span></span><br/><div class="add"><a NAME="E2:121_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ;<br/><a NAME="E3:121_2_1"/><span class="kw">then </span>
(  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="comseq_3.html#K11" title="COMSEQ_3:func.11">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K7" title="RVSUM_4:func.7">Sequel</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 ;<br/><span class="kw">hence </span><a NAME="E4:121_2_1"/>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_3.html#K11" title="COMSEQ_3:func.11">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K7" title="RVSUM_4:func.7">Sequel</a> <font color="Maroon" title="c1">f</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="121_2_2"><span class="kw">suppose </span></a><a NAME="E1:121_2_2"/>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_3.html#K11" title="COMSEQ_3:func.11">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K7" title="RVSUM_4:func.7">Sequel</a> <font color="Maroon" title="c1">f</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">k</font> = <font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/> <a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <font color="Maroon" title="c2">g</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K4" title="SERIES_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T15" onmouseover="rs('series_1/T15')" onmouseout="rh()">SERIES_1:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>

<br/>.= 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T56" onmouseover="rs('afinsq_2/T56')" onmouseout="rh()">AFINSQ_2:56</a></span>
<br/>.= 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c1">f</font>

;<br/><span class="kw">hence </span><a NAME="E4:121_2_2"/>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_3.html#K11" title="COMSEQ_3:func.11">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K7" title="RVSUM_4:func.7">Sequel</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:121"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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