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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">rF</font> being   <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><br/>  for <font color="Olive" title="b2">S</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <font color="Olive" title="b1">rF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">S</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> holds <br/> <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Olive" title="b1">rF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <font color="Olive" title="b2">S</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">rF</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <font color="Maroon" title="c2">rF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">S</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> holds <br/> <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c2">rF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_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="c2">rF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> implies  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c2">rF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">n</font> )</span><br/>





<a NAME="E1:91"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
<font color="Maroon" title="c2">rF</font> is  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="afinsq_2.html#E59"><span class="lab"><font color="Green" title="E48">Lm4</font></span></a></span>;<br/>
<a NAME="E2:91"/>
<font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E3:91"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A2</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:91"/><span class="lab"><font color="Green" title="E56">A3</font></span>: 
<font color="Maroon" title="c2">rF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 ; <a class="txt" 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="c2">rF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">n</font></span><br/>

<a NAME="E5:91"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">rF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/>
<a NAME="E6:91"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">rF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E8:91"/><span class="lab"><font color="Green" title="E58">A5</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">rF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">and </span><br/><a NAME="E9:91"/><span class="lab"><font color="Green" title="E59">A6</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">rF</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">rF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>)
 <span class="kw">and </span><br/><a NAME="E10:91"/><span class="lab"><font color="Green" title="E60">A7</font></span>: 
<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c2">rF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">rF</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#D8" target="_self" title="AFINSQ_2:def.8">Def9</a>, <a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E54">A1</font></span></a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">rF</font> implies <font color="Maroon" title="c4">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 );<br/>
<div><span class="lab"><font color="Green" title="E61">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:91_1"/><span class="lab"><font color="Green" title="E62">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]</span><br/><span class="kw">thus </span><a NAME="E2:91_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:91_1_1"/><span class="lab"><font color="Green" title="E63">A10</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">rF</font>
 ; <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="c4">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span><br/>

<a NAME="E2:91_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A11</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">rF</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/>
<a NAME="E3:91_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A12</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">rF</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E4:91_1_1"/><span class="lab"><font color="Green" title="E66">A13</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> = 
<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">rF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:91"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E2:91_1_1"><span class="lab"><font color="Green" title="E64">A11</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">rF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E1:91_1_1"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1"><span class="lab"><font color="Green" title="E62">A9</font></span></a>, <a class="txt" href="#E3:91_1_1"><span class="lab"><font color="Green" title="E65">A12</font></span></a>, <a class="txt" href="#E4:91_1_1"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a>, <a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</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/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E12:91"/>
<span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></span>;<br/>
<a NAME="E13:91"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A14</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E8:91"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<a NAME="E14:91"/>
 for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E13:91"><span class="lab"><font color="Green" title="E62">A14</font></span></a>, <a class="txt" href="#E11:91"><span class="lab"><font color="Green" title="E61">A8</font></span></a>);<br/></span>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">n</font> = 
<font color="Maroon" title="c4">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:91"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>
<br/>.= 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c2">rF</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T48" target="_self" title="AFINSQ_2:th.48">Th60</a>, <a class="txt" href="#E10:91"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="txt" href="#E6:91"><span class="lab"><font color="Green" title="E57">A4</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
