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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">seq</font> being  <a href="seq_1.html#NM1">Real_Sequence</a><br/> for <font color="Olive">sq</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>   st  <a href="finseq_1.html#K3">len</a> <font color="Olive">sq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">sq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 ) holds <br/>( <font color="Olive">seq</font> is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <font color="Olive">seq</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">sq</font> );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">sq</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>assume </b><a NAME="E1:25_1"/><i><font color="Green">E40</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sq</font> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 holds <br/><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 holds <br/><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 ) )
 ;<br/><a NAME="E2:25_1"/><b>then </b>
<font color="Maroon">sq</font> is    <a href="subset_1.html#M1">Element</a> of 0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/><a NAME="E3:25_1"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">sq</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rvsum_1.html#T109">RVSUM_1:109</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0;<br/><a NAME="E4:25_1"/>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E5:25_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 <b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/><div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/><a NAME="E2:25_1_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:25_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E7:25_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i><a NAME="E8:25_1"/><b>then </b><i><font color="Green">E48</font></i>: 
(  <a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:25_1"/>
<font color="Maroon">seq</font> is <a href="series_1.html#V1">summable</a>
 <b>by </b><i><a class="ref" href="series_1.html#D2">SERIES_1:def 2</a></i>;<br/><b>thus </b><a NAME="E10:25_1"/>
 <a href="series_1.html#K2">Sum</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">sq</font>
 <b>by </b><i>, , <a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:25"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">seq</font> being  <a href="seq_1.html#NM1">Real_Sequence</a><br/> for <font color="Olive">sq</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>   st  <a href="finseq_1.html#K3">len</a> <font color="Olive">sq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> holds <br/><font color="Olive">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">sq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> holds <br/><font color="Olive">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 ) holds <br/>( <font color="Olive">seq</font> is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <font color="Olive">seq</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">sq</font> )
 ;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">sq</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>assume </b><a NAME="E2:25_2"/><i><font color="Green">E51</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 ) )
 ;<br/><a NAME="E3:25_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>then </b><i><font color="Green">E52</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">sq</font> <a href="rfinseq.html#K1">/^</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">sq</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>
<br/>.= 
<font color="Maroon">n</font>
<b>by </b><i/>
;<br/><div><i><font color="Green">E76</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_2_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:25_2_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:25_2_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
<font color="Maroon">sq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">sq</font> <a href="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , , </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_2_3"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:25_2_3"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
0
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><a NAME="E7:25_2"/><b>then </b><i><font color="Green">E87</font></i>: 
( <font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> 1 is <a href="series_1.html#V1">summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">sq</font> <a href="rfinseq.html#K1">/^</a> 1</span>)</span> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E8:25_2"/>
<font color="Maroon">seq</font> is <a href="series_1.html#V1">summable</a>
 <b>by </b><i/>;<br/><b>thus </b> <a href="series_1.html#K2">Sum</a> <font color="Maroon">seq</font> = 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">sq</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">sq</font>
<b>by </b><i>, , </i>
;<br/></div><b>end;</b></div>
<a NAME="E4:25"/><b>then </b><i><font color="Green">E88</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<b>thus </b><a NAME="E5:25"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


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