let vseq be sequence of Complex_l2_Space; :: according to CLVECT_2:def 11 :: thesis: ( not vseq is Cauchy or vseq is convergent )
assume A1: vseq is Cauchy ; :: thesis: vseq is convergent
defpred S₁[ set , set ] means ex i being Element of NAT st
( c₁ = i & ex seqi being Complex_Sequence st
( ( for n being Element of NAT holds seqi . n = (seq_id (vseq . n)) . i ) & seqi is convergent & c₂ = lim seqi ) );
A2: for x being set st x in NAT holds
ex y being set st
( y in COMPLEX & S₁[x,y] )

proof

let x be set ; :: thesis: ( x in NAT implies ex y being set st
( y in COMPLEX & S₁[x,y] ) )
assume x in NAT ; :: thesis: ex y being set st
( y in COMPLEX & S₁[x,y] )
then reconsider i = x as Element of NAT ;
deffunc H₁( set ) -> Element of COMPLEX = (seq_id (vseq . c₁)) . i;
consider seqi being Complex_Sequence such that
A3: for n being Element of NAT holds seqi . n = H₁(n) from COMSEQ_1:sch 1();
take lim seqi ; :: thesis: ( lim seqi in COMPLEX & S₁[x, lim seqi] )

now

let e be Real; :: thesis: ( e > 0 implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
|.((seqi . m) - (seqi . k)).| < e )
assume A4: e > 0 ; :: thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
|.((seqi . m) - (seqi . k)).| < e
thus ex k being Element of NAT st
for m being Element of NAT st k <= m holds
|.((seqi . m) - (seqi . k)).| < e :: thesis: verum

proof

consider k being Element of NAT such that
A5: for n, m being Element of NAT st n >= k & m >= k holds
||.((vseq . n) - (vseq . m)).|| < e by A1, A4, CLVECT_2:58;

now

reconsider e1 = e as Real ;
set vk = vseq . k;
let m be Element of NAT ; :: thesis: ( k <= m implies |.((seqi . m) - (seqi . k)).| < e )
assume A6: k <= m ; :: thesis: |.((seqi . m) - (seqi . k)).| < e
set vm = vseq . m;
||.((vseq . m) - (vseq . k)).|| < e by A5, A6;
then A7: sqrt |.(((vseq . m) - (vseq . k)) .|. ((vseq . m) - (vseq . k))).| < e by CSSPACE:def 15;
set seqmk = seq_id ((vseq . m) - (vseq . k));
A8: sqrt (|.((seqi . m) - (seqi . k)).| * |.((seqi . m) - (seqi . k)).|) = sqrt (|.((seqi . m) - (seqi . k)).| ^2)
.= |.((seqi . m) - (seqi . k)).| by COMPLEX1:46, SQUARE_1:22 ;
seq_id ((vseq . m) - (vseq . k)) = seq_id ((seq_id (vseq . m)) - (seq_id (vseq . k))) by Lm16
.= (seq_id (vseq . m)) + (- (seq_id (vseq . k))) by CSSPACE:1 ;
then (seq_id ((vseq . m) - (vseq . k))) . i = ((seq_id (vseq . m)) . i) + ((- (seq_id (vseq . k))) . i) by VALUED_1:1
.= ((seq_id (vseq . m)) . i) + (- ((seq_id (vseq . k)) . i)) by VALUED_1:8
.= ((seq_id (vseq . m)) . i) - ((seq_id (vseq . k)) . i)
.= (seqi . m) - ((seq_id (vseq . k)) . i) by A3
.= (seqi . m) - (seqi . k) by A3 ;
then |.((seqi . m) - (seqi . k)).| * |.((seqi . m) - (seqi . k)).| = (|.(seq_id ((vseq . m) - (vseq . k))).| . i) * |.((seq_id ((vseq . m) - (vseq . k))) . i).| by VALUED_1:18
.= (|.(seq_id ((vseq . m) - (vseq . k))).| . i) * (|.(seq_id ((vseq . m) - (vseq . k))).| . i) by VALUED_1:18 ;
then (|.(seq_id ((vseq . m) - (vseq . k))).| . i) * (|.((seq_id ((vseq . m) - (vseq . k))) *').| . i) = |.((seqi . m) - (seqi . k)).| * |.((seqi . m) - (seqi . k)).| by Lm19;
then A9: |.((seqi . m) - (seqi . k)).| * |.((seqi . m) - (seqi . k)).| = (|.(seq_id ((vseq . m) - (vseq . k))).| (#) |.((seq_id ((vseq . m) - (vseq . k))) *').|) . i by SEQ_1:8;
0 <= |.((seqi . m) - (seqi . k)).| by COMPLEX1:46;
then A10: 0 <= |.((seqi . m) - (seqi . k)).| * |.((seqi . m) - (seqi . k)).| by XREAL_1:127;
( (seq_id ((vseq . m) - (vseq . k))) (#) ((seq_id ((vseq . m) - (vseq . k))) *') is absolutely_summable & ( for j being Element of NAT holds
( (Re ((seq_id ((vseq . m) - (vseq . k))) (#) ((seq_id ((vseq . m) - (vseq . k))) *'))) . j >= 0 & (Im ((seq_id ((vseq . m) - (vseq . k))) (#) ((seq_id ((vseq . m) - (vseq . k))) *'))) . j = 0 ) ) ) by Lm9, Lm20;
then |.(Sum ((seq_id ((vseq . m) - (vseq . k))) (#) ((seq_id ((vseq . m) - (vseq . k))) *'))).| = Sum |.((seq_id ((vseq . m) - (vseq . k))) (#) ((seq_id ((vseq . m) - (vseq . k))) *')).| by Lm8
.= Sum (|.(seq_id ((vseq . m) - (vseq . k))).| (#) |.((seq_id ((vseq . m) - (vseq . k))) *').|) by COMSEQ_1:46 ;
then A11: sqrt (Sum (|.(seq_id ((vseq . m) - (vseq . k))).| (#) |.((seq_id ((vseq . m) - (vseq . k))) *').|)) < e by A7, Lm18;
A12: for i being Element of NAT holds 0 <= (|.(seq_id ((vseq . m) - (vseq . k))).| (#) |.((seq_id ((vseq . m) - (vseq . k))) *').|) . i

proof

let i be Element of NAT ; :: thesis: 0 <= (|.(seq_id ((vseq . m) - (vseq . k))).| (#) |.((seq_id ((vseq . m) - (vseq . k))) *').|) . i
(|.(seq_id ((vseq . m) - (vseq . k))).| (#) |.((seq_id ((vseq . m) - (vseq . k))) *').|) . i = (|.(seq_id ((vseq . m) - (vseq . k))).| . i) * (|.((seq_id ((vseq . m) - (vseq . k))) *').| . i) by SEQ_1:8
.= (|.(seq_id ((vseq . m) - (vseq . k))).| . i) * (|.(seq_id ((vseq . m) - (vseq . k))).| . i) by Lm19 ;
hence 0 <= (|.(seq_id ((vseq . m) - (vseq . k))).| (#) |.((seq_id ((vseq . m) - (vseq . k))) *').|) . i by XREAL_1:63; :: thesis: verum

end;

|.(seq_id ((vseq . m) - (vseq . k))).| (#) |.(seq_id ((vseq . m) - (vseq . k))).| is summable by CSSPACE:def 11, CSSPACE:def 18;
then A13: |.(seq_id ((vseq . m) - (vseq . k))).| (#) |.((seq_id ((vseq . m) - (vseq . k))) *').| is summable by Lm19;
then A14: 0 <= Sum (|.(seq_id ((vseq . m) - (vseq . k))).| (#) |.((seq_id ((vseq . m) - (vseq . k))) *').|) by A12, SERIES_1:18;
then 0 <= sqrt (Sum (|.(seq_id ((vseq . m) - (vseq . k))).| (#) |.((seq_id ((vseq . m) - (vseq . k))) *').|)) by SQUARE_1:def 2;
then (sqrt (Sum (|.(seq_id ((vseq . m) - (vseq . k))).| (#) |.((seq_id ((vseq . m) - (vseq . k))) *').|))) ^2 < e1 ^2 by A11, SQUARE_1:16;
then A15: Sum (|.(seq_id ((vseq . m) - (vseq . k))).| (#) |.((seq_id ((vseq . m) - (vseq . k))) *').|) < e * e by A14, SQUARE_1:def 2;
(|.(seq_id ((vseq . m) - (vseq . k))).| (#) |.((seq_id ((vseq . m) - (vseq . k))) *').|) . i <= Sum (|.(seq_id ((vseq . m) - (vseq . k))).| (#) |.((seq_id ((vseq . m) - (vseq . k))) *').|) by A13, A12, RSSPACE2:3;
then A16: |.((seqi . m) - (seqi . k)).| * |.((seqi . m) - (seqi . k)).| < e * e by A15, A9, XXREAL_0:2;
sqrt (e * e) = sqrt (e1 ^2)
.= e by A4, SQUARE_1:22 ;
hence |.((seqi . m) - (seqi . k)).| < e by A10, A16, A8, SQUARE_1:27; :: thesis: verum

end;

hence ex k being Element of NAT st
for m being Element of NAT st k <= m holds
|.((seqi . m) - (seqi . k)).| < e ; :: thesis: verum

end;

end;

then seqi is convergent by COMSEQ_3:46;
hence ( lim seqi in COMPLEX & S₁[x, lim seqi] ) by A3; :: thesis: verum

end;

consider f being Function of NAT,COMPLEX such that
A17: for x being set st x in NAT holds
S₁[x,f . x] from FUNCT_2:sch 1(A2);
reconsider tseq = f as Complex_Sequence ;
set seqt = seq_id tseq;

A18: now

let i be Element of NAT ; :: thesis: ex seqi being Complex_Sequence st
( ( for n being Element of NAT holds seqi . n = (seq_id (vseq . n)) . i ) & seqi is convergent & tseq . i = lim seqi )
reconsider x = i as set ;
ex i0 being Element of NAT st
( x = i0 & ex seqi being Complex_Sequence st
( ( for n being Element of NAT holds seqi . n = (seq_id (vseq . n)) . i0 ) & seqi is convergent & f . x = lim seqi ) ) by A17;
hence ex seqi being Complex_Sequence st
( ( for n being Element of NAT holds seqi . n = (seq_id (vseq . n)) . i ) & seqi is convergent & tseq . i = lim seqi ) ; :: thesis: verum

end;

A19: for e being Real st e > 0 holds
ex k being Element of NAT st
for n being Element of NAT st n >= k holds
( |.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).| is summable & |.(Sum (((seq_id tseq) - (seq_id (vseq . n))) (#) (((seq_id tseq) - (seq_id (vseq . n))) *'))).| < e * e )

proof

let e1 be Real; :: thesis: ( e1 > 0 implies ex k being Element of NAT st
for n being Element of NAT st n >= k holds
( |.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).| is summable & |.(Sum (((seq_id tseq) - (seq_id (vseq . n))) (#) (((seq_id tseq) - (seq_id (vseq . n))) *'))).| < e1 * e1 ) )
assume A20: e1 > 0 ; :: thesis: ex k being Element of NAT st
for n being Element of NAT st n >= k holds
( |.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).| is summable & |.(Sum (((seq_id tseq) - (seq_id (vseq . n))) (#) (((seq_id tseq) - (seq_id (vseq . n))) *'))).| < e1 * e1 )
set e = e1 / 2;
A21: e1 / 2 > 0 by A20, XREAL_1:215;
then consider k being Element of NAT such that
A22: for n, m being Element of NAT st n >= k & m >= k holds
||.((vseq . n) - (vseq . m)).|| < e1 / 2 by A1, CLVECT_2:58;
e1 / 2 < e1 by A20, XREAL_1:216;
then A23: (e1 / 2) * (e1 / 2) < e1 * e1 by A21, XREAL_1:97;
A24: for m, n being Element of NAT st n >= k & m >= k holds
( |.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).| is summable & Sum (|.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).|) < (e1 / 2) * (e1 / 2) & ( for i being Element of NAT holds 0 <= (|.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).|) . i ) )

proof

let m, n be Element of NAT ; :: thesis: ( n >= k & m >= k implies ( |.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).| is summable & Sum (|.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).|) < (e1 / 2) * (e1 / 2) & ( for i being Element of NAT holds 0 <= (|.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).|) . i ) ) )
assume ( n >= k & m >= k ) ; :: thesis: ( |.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).| is summable & Sum (|.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).|) < (e1 / 2) * (e1 / 2) & ( for i being Element of NAT holds 0 <= (|.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).|) . i ) )
then ||.((vseq . n) - (vseq . m)).|| < e1 / 2 by A22;
then A25: sqrt |.(((vseq . n) - (vseq . m)) .|. ((vseq . n) - (vseq . m))).| < e1 / 2 by CSSPACE:def 15;
( (seq_id ((vseq . n) - (vseq . m))) (#) ((seq_id ((vseq . n) - (vseq . m))) *') is absolutely_summable & ( for j being Element of NAT holds
( (Re ((seq_id ((vseq . n) - (vseq . m))) (#) ((seq_id ((vseq . n) - (vseq . m))) *'))) . j >= 0 & (Im ((seq_id ((vseq . n) - (vseq . m))) (#) ((seq_id ((vseq . n) - (vseq . m))) *'))) . j = 0 ) ) ) by Lm9, Lm20;
then |.(Sum ((seq_id ((vseq . n) - (vseq . m))) (#) ((seq_id ((vseq . n) - (vseq . m))) *'))).| = Sum |.((seq_id ((vseq . n) - (vseq . m))) (#) ((seq_id ((vseq . n) - (vseq . m))) *')).| by Lm8
.= Sum (|.(seq_id ((vseq . n) - (vseq . m))).| (#) |.((seq_id ((vseq . n) - (vseq . m))) *').|) by COMSEQ_1:46
.= Sum (|.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).|) by Lm19
.= Sum |.((seq_id ((vseq . n) - (vseq . m))) (#) (seq_id ((vseq . n) - (vseq . m)))).| by COMSEQ_1:46 ;
then A26: sqrt (Sum |.((seq_id ((vseq . n) - (vseq . m))) (#) (seq_id ((vseq . n) - (vseq . m)))).|) < e1 / 2 by A25, Lm18;
A27: for i being Element of NAT holds 0 <= (|.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).|) . i

proof

let i be Element of NAT ; :: thesis: 0 <= (|.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).|) . i
(|.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).|) . i = |.((seq_id ((vseq . n) - (vseq . m))) (#) (seq_id ((vseq . n) - (vseq . m)))).| . i by COMSEQ_1:46
.= |.(((seq_id ((vseq . n) - (vseq . m))) (#) (seq_id ((vseq . n) - (vseq . m)))) . i).| by VALUED_1:18 ;
hence 0 <= (|.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).|) . i by COMPLEX1:46; :: thesis: verum

end;

|.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).| is summable by CSSPACE:def 11, CSSPACE:def 18;
then 0 <= Sum (|.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).|) by A27, SERIES_1:18;
then A28: 0 <= Sum |.((seq_id ((vseq . n) - (vseq . m))) (#) (seq_id ((vseq . n) - (vseq . m)))).| by COMSEQ_1:46;
then 0 <= sqrt (Sum |.((seq_id ((vseq . n) - (vseq . m))) (#) (seq_id ((vseq . n) - (vseq . m)))).|) by SQUARE_1:def 2;
then (sqrt (Sum |.((seq_id ((vseq . n) - (vseq . m))) (#) (seq_id ((vseq . n) - (vseq . m)))).|)) ^2 < (e1 / 2) ^2 by A26, SQUARE_1:16;
then Sum |.((seq_id ((vseq . n) - (vseq . m))) (#) (seq_id ((vseq . n) - (vseq . m)))).| < (e1 / 2) * (e1 / 2) by A28, SQUARE_1:def 2;
hence ( |.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).| is summable & Sum (|.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).|) < (e1 / 2) * (e1 / 2) & ( for i being Element of NAT holds 0 <= (|.(seq_id ((vseq . n) - (vseq . m))).| (#) |.(seq_id ((vseq . n) - (vseq . m))).|) . i ) ) by A27, COMSEQ_1:46, CSSPACE:def 11, CSSPACE:def 18; :: thesis: verum

end;

A29: for n, i being Element of NAT
for rseq being Real_Sequence st ( for m being Element of NAT holds rseq . m = (Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . i ) holds
( rseq is convergent & lim rseq = (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . i )

proof

let n be Element of NAT ; :: thesis: for i being Element of NAT
for rseq being Real_Sequence st ( for m being Element of NAT holds rseq . m = (Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . i ) holds
( rseq is convergent & lim rseq = (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . i )
defpred S₂[ Element of NAT ] means for rseq being Real_Sequence st ( for m being Element of NAT holds rseq . m = (Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . c₁ ) holds
( rseq is convergent & lim rseq = (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . c₁ );
A30: for m, k being Element of NAT holds seq_id ((vseq . m) - (vseq . k)) = (seq_id (vseq . m)) - (seq_id (vseq . k))

proof

let m, k be Element of NAT ; :: thesis: seq_id ((vseq . m) - (vseq . k)) = (seq_id (vseq . m)) - (seq_id (vseq . k))
thus seq_id ((vseq . m) - (vseq . k)) = seq_id ((seq_id (vseq . m)) - (seq_id (vseq . k))) by Lm16
.= (seq_id (vseq . m)) - (seq_id (vseq . k)) by CSSPACE:1 ; :: thesis: verum

end;

now

let i be Element of NAT ; :: thesis: ( ( for rseq being Real_Sequence st ( for m being Element of NAT holds rseq . m = (Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . i ) holds
( rseq is convergent & lim rseq = (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . i ) ) implies for rseq being Real_Sequence st ( for m being Element of NAT holds rseq . m = (Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . (i + 1) ) holds
( rseq is convergent & lim rseq = (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . (i + 1) ) )
assume A31: for rseq being Real_Sequence st ( for m being Element of NAT holds rseq . m = (Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . i ) holds
( rseq is convergent & lim rseq = (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . i ) ; :: thesis: for rseq being Real_Sequence st ( for m being Element of NAT holds rseq . m = (Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . (i + 1) ) holds
( rseq is convergent & lim rseq = (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . (i + 1) )
thus for rseq being Real_Sequence st ( for m being Element of NAT holds rseq . m = (Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . (i + 1) ) holds
( rseq is convergent & lim rseq = (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . (i + 1) ) :: thesis: verum

proof

deffunc H₁( Element of NAT ) -> Element of REAL = (Partial_Sums (|.(seq_id ((vseq . c₁) - (vseq . n))).| (#) |.(seq_id ((vseq . c₁) - (vseq . n))).|)) . i;
consider rseqb being Real_Sequence such that
A32: for m being Element of NAT holds rseqb . m = H₁(m) from SEQ_1:sch 1();
consider seq0 being Complex_Sequence such that
A33: for m being Element of NAT holds seq0 . m = (seq_id (vseq . m)) . (i + 1) and
A34: seq0 is convergent and
A35: tseq . (i + 1) = lim seq0 by A18;
let rseq be Real_Sequence; :: thesis: ( ( for m being Element of NAT holds rseq . m = (Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . (i + 1) ) implies ( rseq is convergent & lim rseq = (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . (i + 1) ) )
assume A36: for m being Element of NAT holds rseq . m = (Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . (i + 1) ; :: thesis: ( rseq is convergent & lim rseq = (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . (i + 1) )

A37: now

let m be Element of NAT ; :: thesis: rseq . m = (|.((seq0 . m) - ((seq_id (vseq . n)) . (i + 1))).| * |.((seq0 . m) - ((seq_id (vseq . n)) . (i + 1))).|) + (rseqb . m)
thus rseq . m = (Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . (i + 1) by A36
.= ((|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|) . (i + 1)) + ((Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . i) by SERIES_1:def 1
.= ((|.((seq_id (vseq . m)) - (seq_id (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|) . (i + 1)) + ((Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . i) by A30
.= ((|.((seq_id (vseq . m)) - (seq_id (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|) . (i + 1)) + (rseqb . m) by A32
.= ((|.((seq_id (vseq . m)) - (seq_id (vseq . n))).| (#) |.((seq_id (vseq . m)) - (seq_id (vseq . n))).|) . (i + 1)) + (rseqb . m) by A30
.= ((|.((seq_id (vseq . m)) + (- (seq_id (vseq . n)))).| . (i + 1)) * (|.((seq_id (vseq . m)) - (seq_id (vseq . n))).| . (i + 1))) + (rseqb . m) by SEQ_1:8
.= (|.(((seq_id (vseq . m)) + (- (seq_id (vseq . n)))) . (i + 1)).| * (|.((seq_id (vseq . m)) + (- (seq_id (vseq . n)))).| . (i + 1))) + (rseqb . m) by VALUED_1:18
.= (|.(((seq_id (vseq . m)) + (- (seq_id (vseq . n)))) . (i + 1)).| * |.(((seq_id (vseq . m)) + (- (seq_id (vseq . n)))) . (i + 1)).|) + (rseqb . m) by VALUED_1:18
.= (|.(((seq_id (vseq . m)) . (i + 1)) + ((- (seq_id (vseq . n))) . (i + 1))).| * |.(((seq_id (vseq . m)) + (- (seq_id (vseq . n)))) . (i + 1)).|) + (rseqb . m) by VALUED_1:1
.= (|.(((seq_id (vseq . m)) . (i + 1)) + ((- (seq_id (vseq . n))) . (i + 1))).| * |.(((seq_id (vseq . m)) . (i + 1)) + ((- (seq_id (vseq . n))) . (i + 1))).|) + (rseqb . m) by VALUED_1:1
.= (|.(((seq_id (vseq . m)) . (i + 1)) + (- ((seq_id (vseq . n)) . (i + 1)))).| * |.(((seq_id (vseq . m)) . (i + 1)) + ((- (seq_id (vseq . n))) . (i + 1))).|) + (rseqb . m) by VALUED_1:8
.= (|.(((seq_id (vseq . m)) . (i + 1)) - ((seq_id (vseq . n)) . (i + 1))).| * |.(((seq_id (vseq . m)) . (i + 1)) + (- ((seq_id (vseq . n)) . (i + 1)))).|) + (rseqb . m) by VALUED_1:8
.= (|.((seq0 . m) - ((seq_id (vseq . n)) . (i + 1))).| * |.(((seq_id (vseq . m)) . (i + 1)) - ((seq_id (vseq . n)) . (i + 1))).|) + (rseqb . m) by A33
.= (|.((seq0 . m) - ((seq_id (vseq . n)) . (i + 1))).| * |.((seq0 . m) - ((seq_id (vseq . n)) . (i + 1))).|) + (rseqb . m) by A33 ; :: thesis: verum

end;

A38: rseqb is convergent by A31, A32;
then lim rseq = (|.((lim seq0) - ((seq_id (vseq . n)) . (i + 1))).| * |.((lim seq0) - ((seq_id (vseq . n)) . (i + 1))).|) + (lim rseqb) by A34, A37, Lm24
.= (|.((tseq . (i + 1)) + ((- (seq_id (vseq . n))) . (i + 1))).| * |.((tseq . (i + 1)) + (- ((seq_id (vseq . n)) . (i + 1)))).|) + (lim rseqb) by A35, VALUED_1:8
.= (|.((tseq . (i + 1)) + ((- (seq_id (vseq . n))) . (i + 1))).| * |.((tseq . (i + 1)) + ((- (seq_id (vseq . n))) . (i + 1))).|) + (lim rseqb) by VALUED_1:8
.= (|.((tseq + (- (seq_id (vseq . n)))) . (i + 1)).| * |.((tseq . (i + 1)) + ((- (seq_id (vseq . n))) . (i + 1))).|) + (lim rseqb) by VALUED_1:1
.= (|.((tseq + (- (seq_id (vseq . n)))) . (i + 1)).| * |.((tseq + (- (seq_id (vseq . n)))) . (i + 1)).|) + (lim rseqb) by VALUED_1:1
.= ((|.(tseq + (- (seq_id (vseq . n)))).| . (i + 1)) * |.((tseq + (- (seq_id (vseq . n)))) . (i + 1)).|) + (lim rseqb) by VALUED_1:18
.= ((|.(tseq - (seq_id (vseq . n))).| . (i + 1)) * (|.(tseq + (- (seq_id (vseq . n)))).| . (i + 1))) + (lim rseqb) by VALUED_1:18
.= ((|.(tseq - (seq_id (vseq . n))).| (#) |.(tseq - (seq_id (vseq . n))).|) . (i + 1)) + (lim rseqb) by SEQ_1:8
.= ((|.(tseq - (seq_id (vseq . n))).| (#) |.(tseq - (seq_id (vseq . n))).|) . (i + 1)) + ((Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . i) by A31, A32
.= ((|.(tseq - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|) . (i + 1)) + ((Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . i) by CSSPACE:1
.= ((|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|) . (i + 1)) + ((Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . i) by CSSPACE:1
.= (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . (i + 1) by SERIES_1:def 1 ;
hence ( rseq is convergent & lim rseq = (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . (i + 1) ) by A38, A34, A37, Lm24; :: thesis: verum

end;

end;

then A39: for i being Element of NAT st S₂[i] holds
S₂[i + 1] ;

now

let rseq be Real_Sequence; :: thesis: ( ( for m being Element of NAT holds rseq . m = (Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . 0 ) implies ( rseq is convergent & lim rseq = (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . 0 ) )
assume A40: for m being Element of NAT holds rseq . m = (Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . 0 ; :: thesis: ( rseq is convergent & lim rseq = (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . 0 )
thus ( rseq is convergent & lim rseq = (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . 0 ) :: thesis: verum

proof

consider rseq0 being Complex_Sequence such that
A41: for m being Element of NAT holds rseq0 . m = (seq_id (vseq . m)) . 0 and
A42: rseq0 is convergent and
A43: tseq . 0 = lim rseq0 by A18;

A44: now

let m be Element of NAT ; :: thesis: rseq . m = |.((rseq0 . m) - ((seq_id (vseq . n)) . 0)).| * |.((rseq0 . m) - ((seq_id (vseq . n)) . 0)).|
thus rseq . m = (Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . 0 by A40
.= (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|) . 0 by SERIES_1:def 1
.= (|.((seq_id (vseq . m)) - (seq_id (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|) . 0 by A30
.= (|.((seq_id (vseq . m)) - (seq_id (vseq . n))).| (#) |.((seq_id (vseq . m)) - (seq_id (vseq . n))).|) . 0 by A30
.= (|.((seq_id (vseq . m)) + (- (seq_id (vseq . n)))).| . 0) * (|.((seq_id (vseq . m)) - (seq_id (vseq . n))).| . 0) by SEQ_1:8
.= |.(((seq_id (vseq . m)) + (- (seq_id (vseq . n)))) . 0).| * (|.((seq_id (vseq . m)) + (- (seq_id (vseq . n)))).| . 0) by VALUED_1:18
.= |.(((seq_id (vseq . m)) + (- (seq_id (vseq . n)))) . 0).| * |.(((seq_id (vseq . m)) + (- (seq_id (vseq . n)))) . 0).| by VALUED_1:18
.= |.(((seq_id (vseq . m)) . 0) + ((- (seq_id (vseq . n))) . 0)).| * |.(((seq_id (vseq . m)) + (- (seq_id (vseq . n)))) . 0).| by VALUED_1:1
.= |.(((seq_id (vseq . m)) . 0) + ((- (seq_id (vseq . n))) . 0)).| * |.(((seq_id (vseq . m)) . 0) + ((- (seq_id (vseq . n))) . 0)).| by VALUED_1:1
.= |.(((seq_id (vseq . m)) . 0) + (- ((seq_id (vseq . n)) . 0))).| * |.(((seq_id (vseq . m)) . 0) + ((- (seq_id (vseq . n))) . 0)).| by VALUED_1:8
.= |.(((seq_id (vseq . m)) . 0) + (- ((seq_id (vseq . n)) . 0))).| * |.(((seq_id (vseq . m)) . 0) + (- ((seq_id (vseq . n)) . 0))).| by VALUED_1:8
.= |.((rseq0 . m) - ((seq_id (vseq . n)) . 0)).| * |.(((seq_id (vseq . m)) . 0) - ((seq_id (vseq . n)) . 0)).| by A41
.= |.((rseq0 . m) - ((seq_id (vseq . n)) . 0)).| * |.((rseq0 . m) - ((seq_id (vseq . n)) . 0)).| by A41 ; :: thesis: verum

end;

then lim rseq = |.((tseq . 0) - ((seq_id (vseq . n)) . 0)).| * |.((tseq . 0) - ((seq_id (vseq . n)) . 0)).| by A42, A43, Lm23
.= |.((tseq . 0) + ((- (seq_id (vseq . n))) . 0)).| * |.((tseq . 0) + (- ((seq_id (vseq . n)) . 0))).| by VALUED_1:8
.= |.((tseq . 0) + ((- (seq_id (vseq . n))) . 0)).| * |.((tseq . 0) + ((- (seq_id (vseq . n))) . 0)).| by VALUED_1:8
.= |.((tseq + (- (seq_id (vseq . n)))) . 0).| * |.((tseq . 0) + ((- (seq_id (vseq . n))) . 0)).| by VALUED_1:1
.= |.((tseq - (seq_id (vseq . n))) . 0).| * |.((tseq + (- (seq_id (vseq . n)))) . 0).| by VALUED_1:1
.= (|.(tseq - (seq_id (vseq . n))).| . 0) * |.((tseq - (seq_id (vseq . n))) . 0).| by VALUED_1:18
.= (|.(tseq - (seq_id (vseq . n))).| . 0) * (|.(tseq - (seq_id (vseq . n))).| . 0) by VALUED_1:18
.= (|.(tseq - (seq_id (vseq . n))).| (#) |.(tseq - (seq_id (vseq . n))).|) . 0 by SEQ_1:8
.= (Partial_Sums (|.(tseq - (seq_id (vseq . n))).| (#) |.(tseq - (seq_id (vseq . n))).|)) . 0 by SERIES_1:def 1
.= (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.(tseq - (seq_id (vseq . n))).|)) . 0 by CSSPACE:1
.= (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . 0 by CSSPACE:1 ;
hence ( rseq is convergent & lim rseq = (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . 0 ) by A42, A44, Lm23; :: thesis: verum

end;

end;

then A45: S₂[ 0 ] ;
for i being Element of NAT holds S₂[i] from NAT_1:sch 1(A45, A39);
hence for i being Element of NAT
for rseq being Real_Sequence st ( for m being Element of NAT holds rseq . m = (Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . i ) holds
( rseq is convergent & lim rseq = (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . i ) ; :: thesis: verum

end;

for n being Element of NAT st n >= k holds
( |.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).| is summable & |.(Sum (((seq_id tseq) - (seq_id (vseq . n))) (#) (((seq_id tseq) - (seq_id (vseq . n))) *'))).| < e1 * e1 )

proof

let n be Element of NAT ; :: thesis: ( n >= k implies ( |.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).| is summable & |.(Sum (((seq_id tseq) - (seq_id (vseq . n))) (#) (((seq_id tseq) - (seq_id (vseq . n))) *'))).| < e1 * e1 ) )
assume A46: n >= k ; :: thesis: ( |.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).| is summable & |.(Sum (((seq_id tseq) - (seq_id (vseq . n))) (#) (((seq_id tseq) - (seq_id (vseq . n))) *'))).| < e1 * e1 )
A47: for i being Element of NAT st 0 <= i holds
(Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . i <= (e1 / 2) * (e1 / 2)

proof

let i be Element of NAT ; :: thesis: ( 0 <= i implies (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . i <= (e1 / 2) * (e1 / 2) )
assume 0 <= i ; :: thesis: (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . i <= (e1 / 2) * (e1 / 2)
deffunc H₁( Element of NAT ) -> Element of REAL = (Partial_Sums (|.(seq_id ((vseq . c₁) - (vseq . n))).| (#) |.(seq_id ((vseq . c₁) - (vseq . n))).|)) . i;
consider rseq being Real_Sequence such that
A48: for m being Element of NAT holds rseq . m = H₁(m) from SEQ_1:sch 1();
A49: for m being Element of NAT st m >= k holds
rseq . m <= (e1 / 2) * (e1 / 2)

proof

let m be Element of NAT ; :: thesis: ( m >= k implies rseq . m <= (e1 / 2) * (e1 / 2) )
assume A50: m >= k ; :: thesis: rseq . m <= (e1 / 2) * (e1 / 2)
( |.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).| is summable & ( for i being Element of NAT holds 0 <= (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|) . i ) ) by A24, A46, A50;
then A51: (Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . i <= Sum (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|) by RSSPACE2:3;
A52: rseq . m = (Partial_Sums (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|)) . i by A48;
Sum (|.(seq_id ((vseq . m) - (vseq . n))).| (#) |.(seq_id ((vseq . m) - (vseq . n))).|) < (e1 / 2) * (e1 / 2) by A24, A46, A50;
hence rseq . m <= (e1 / 2) * (e1 / 2) by A52, A51, XXREAL_0:2; :: thesis: verum

end;

( rseq is convergent & lim rseq = (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . i ) by A29, A48;
hence (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . i <= (e1 / 2) * (e1 / 2) by A49, RSSPACE2:5; :: thesis: verum

end;

now

let i be Element of NAT ; :: thesis: (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . i < e1 * e1
(Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . i <= (e1 / 2) * (e1 / 2) by A47, NAT_1:2;
hence (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) . i < e1 * e1 by A23, XXREAL_0:2; :: thesis: verum

end;

then A53: Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|) is bounded_above by SEQ_2:def 3;
A54: for i being Element of NAT holds 0 <= (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|) . i

proof

let i be Element of NAT ; :: thesis: 0 <= (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|) . i
(|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|) . i = (|.((seq_id tseq) - (seq_id (vseq . n))).| . i) * (|.((seq_id tseq) - (seq_id (vseq . n))).| . i) by SEQ_1:8;
hence 0 <= (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|) . i by XREAL_1:63; :: thesis: verum

end;

then A55: |.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).| is summable by A53, SERIES_1:17;
then Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|) is convergent by SERIES_1:def 2;
then ( Sum (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|) = lim (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) & lim (Partial_Sums (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|)) <= (e1 / 2) * (e1 / 2) ) by A47, RSSPACE2:5, SERIES_1:def 3;
then A56: Sum (|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).|) < e1 * e1 by A23, XXREAL_0:2;
|.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.(((seq_id tseq) - (seq_id (vseq . n))) *').| is summable by A55, Lm19;
then |.(((seq_id tseq) - (seq_id (vseq . n))) (#) (((seq_id tseq) - (seq_id (vseq . n))) *')).| is summable by COMSEQ_1:46;
then ((seq_id tseq) - (seq_id (vseq . n))) (#) (((seq_id tseq) - (seq_id (vseq . n))) *') is absolutely_summable by COMSEQ_3:def 9;
then A57: |.(Sum (((seq_id tseq) - (seq_id (vseq . n))) (#) (((seq_id tseq) - (seq_id (vseq . n))) *'))).| <= Sum |.(((seq_id tseq) - (seq_id (vseq . n))) (#) (((seq_id tseq) - (seq_id (vseq . n))) *')).| by Lm7;
|.((seq_id tseq) - (seq_id (vseq . n))).| = |.(((seq_id tseq) - (seq_id (vseq . n))) *').| by Lm19;
then Sum |.(((seq_id tseq) - (seq_id (vseq . n))) (#) (((seq_id tseq) - (seq_id (vseq . n))) *')).| < e1 * e1 by A56, COMSEQ_1:46;
hence ( |.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).| is summable & |.(Sum (((seq_id tseq) - (seq_id (vseq . n))) (#) (((seq_id tseq) - (seq_id (vseq . n))) *'))).| < e1 * e1 ) by A54, A53, A57, SERIES_1:17, XXREAL_0:2; :: thesis: verum

end;

hence ex k being Element of NAT st
for n being Element of NAT st n >= k holds
( |.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).| is summable & |.(Sum (((seq_id tseq) - (seq_id (vseq . n))) (#) (((seq_id tseq) - (seq_id (vseq . n))) *'))).| < e1 * e1 ) ; :: thesis: verum

end;

A58: |.(seq_id tseq).| (#) |.(seq_id tseq).| is summable

proof

set d = |.(seq_id tseq).| (#) |.(seq_id tseq).|;
A59: for i being Element of NAT holds 0 <= (|.(seq_id tseq).| (#) |.(seq_id tseq).|) . i

proof

let i be Element of NAT ; :: thesis: 0 <= (|.(seq_id tseq).| (#) |.(seq_id tseq).|) . i
(|.(seq_id tseq).| (#) |.(seq_id tseq).|) . i = (|.(seq_id tseq).| . i) * (|.(seq_id tseq).| . i) by SEQ_1:8;
hence 0 <= (|.(seq_id tseq).| (#) |.(seq_id tseq).|) . i by XREAL_1:63; :: thesis: verum

end;

consider m being Element of NAT such that
A60: for n being Element of NAT st n >= m holds
( |.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).| is summable & |.(Sum (((seq_id tseq) - (seq_id (vseq . n))) (#) (((seq_id tseq) - (seq_id (vseq . n))) *'))).| < 1 * 1 ) by A19;
set b = |.(seq_id (vseq . m)).| (#) |.(seq_id (vseq . m)).|;
set a = |.((seq_id tseq) - (seq_id (vseq . m))).| (#) |.((seq_id tseq) - (seq_id (vseq . m))).|;
|.(seq_id (vseq . m)).| (#) |.(seq_id (vseq . m)).| is summable by CSSPACE:def 11, CSSPACE:def 18;
then A61: 2 (#) (|.(seq_id (vseq . m)).| (#) |.(seq_id (vseq . m)).|) is summable by SERIES_1:10;
A62: for i being Element of NAT holds (|.(seq_id tseq).| (#) |.(seq_id tseq).|) . i <= ((2 (#) (|.((seq_id tseq) - (seq_id (vseq . m))).| (#) |.((seq_id tseq) - (seq_id (vseq . m))).|)) + (2 (#) (|.(seq_id (vseq . m)).| (#) |.(seq_id (vseq . m)).|))) . i

proof

let i be Element of NAT ; :: thesis: (|.(seq_id tseq).| (#) |.(seq_id tseq).|) . i <= ((2 (#) (|.((seq_id tseq) - (seq_id (vseq . m))).| (#) |.((seq_id tseq) - (seq_id (vseq . m))).|)) + (2 (#) (|.(seq_id (vseq . m)).| (#) |.(seq_id (vseq . m)).|))) . i
A63: ((2 (#) (|.((seq_id tseq) - (seq_id (vseq . m))).| (#) |.((seq_id tseq) - (seq_id (vseq . m))).|)) + (2 (#) (|.(seq_id (vseq . m)).| (#) |.(seq_id (vseq . m)).|))) . i = ((2 (#) (|.((seq_id tseq) - (seq_id (vseq . m))).| (#) |.((seq_id tseq) - (seq_id (vseq . m))).|)) . i) + ((2 (#) (|.(seq_id (vseq . m)).| (#) |.(seq_id (vseq . m)).|)) . i) by SEQ_1:7
.= (2 * ((|.((seq_id tseq) - (seq_id (vseq . m))).| (#) |.((seq_id tseq) - (seq_id (vseq . m))).|) . i)) + ((2 (#) (|.(seq_id (vseq . m)).| (#) |.(seq_id (vseq . m)).|)) . i) by SEQ_1:9
.= (2 * ((|.((seq_id tseq) - (seq_id (vseq . m))).| (#) |.((seq_id tseq) - (seq_id (vseq . m))).|) . i)) + (2 * ((|.(seq_id (vseq . m)).| (#) |.(seq_id (vseq . m)).|) . i)) by SEQ_1:9 ;
set y = (seq_id (vseq . m)) . i;
set x = (seq_id tseq) . i;
A64: 2 * ((|.(seq_id (vseq . m)).| (#) |.(seq_id (vseq . m)).|) . i) = 2 * ((|.(seq_id (vseq . m)).| . i) * (|.(seq_id (vseq . m)).| . i)) by SEQ_1:8
.= 2 * (|.((seq_id (vseq . m)) . i).| * (|.(seq_id (vseq . m)).| . i)) by VALUED_1:18
.= 2 * (|.((seq_id (vseq . m)) . i).| * |.((seq_id (vseq . m)) . i).|) by VALUED_1:18
.= (2 * |.((seq_id (vseq . m)) . i).|) * |.((seq_id (vseq . m)) . i).| ;
|.((seq_id tseq) - (seq_id (vseq . m))).| . i = |.(((seq_id tseq) + (- (seq_id (vseq . m)))) . i).| by VALUED_1:18
.= |.(((seq_id tseq) . i) + ((- (seq_id (vseq . m))) . i)).| by VALUED_1:1
.= |.(((seq_id tseq) . i) + (- ((seq_id (vseq . m)) . i))).| by VALUED_1:8
.= |.(((seq_id tseq) . i) - ((seq_id (vseq . m)) . i)).| ;
then (|.((seq_id tseq) - (seq_id (vseq . m))).| (#) |.((seq_id tseq) - (seq_id (vseq . m))).|) . i = |.(((seq_id tseq) . i) - ((seq_id (vseq . m)) . i)).| * |.(((seq_id tseq) . i) - ((seq_id (vseq . m)) . i)).| by SEQ_1:8;
then A65: 2 * ((|.((seq_id tseq) - (seq_id (vseq . m))).| (#) |.((seq_id tseq) - (seq_id (vseq . m))).|) . i) = (2 * |.(((seq_id tseq) . i) - ((seq_id (vseq . m)) . i)).|) * |.(((seq_id tseq) . i) - ((seq_id (vseq . m)) . i)).| ;
(|.(seq_id tseq).| (#) |.(seq_id tseq).|) . i = (|.(seq_id tseq).| . i) * (|.(seq_id tseq).| . i) by SEQ_1:8
.= |.((seq_id tseq) . i).| * (|.(seq_id tseq).| . i) by VALUED_1:18
.= |.((seq_id tseq) . i).| * |.((seq_id tseq) . i).| by VALUED_1:18 ;
hence (|.(seq_id tseq).| (#) |.(seq_id tseq).|) . i <= ((2 (#) (|.((seq_id tseq) - (seq_id (vseq . m))).| (#) |.((seq_id tseq) - (seq_id (vseq . m))).|)) + (2 (#) (|.(seq_id (vseq . m)).| (#) |.(seq_id (vseq . m)).|))) . i by A63, A65, A64, Lm22; :: thesis: verum

end;

|.((seq_id tseq) - (seq_id (vseq . m))).| (#) |.((seq_id tseq) - (seq_id (vseq . m))).| is summable by A60;
then 2 (#) (|.((seq_id tseq) - (seq_id (vseq . m))).| (#) |.((seq_id tseq) - (seq_id (vseq . m))).|) is summable by SERIES_1:10;
then (2 (#) (|.((seq_id tseq) - (seq_id (vseq . m))).| (#) |.((seq_id tseq) - (seq_id (vseq . m))).|)) + (2 (#) (|.(seq_id (vseq . m)).| (#) |.(seq_id (vseq . m)).|)) is summable by A61, SERIES_1:7;
hence |.(seq_id tseq).| (#) |.(seq_id tseq).| is summable by A59, A62, SERIES_1:20; :: thesis: verum

end;

tseq in the_set_of_ComplexSequences by CSSPACE:def 1;
then reconsider tv = tseq as Point of Complex_l2_Space by A58, CSSPACE:def 11, CSSPACE:def 18;
for e being Real st e > 0 holds
ex m being Element of NAT st
for n being Element of NAT st n >= m holds
||.((vseq . n) - tv).|| < e

proof

let e be Real; :: thesis: ( e > 0 implies ex m being Element of NAT st
for n being Element of NAT st n >= m holds
||.((vseq . n) - tv).|| < e )
assume A66: e > 0 ; :: thesis: ex m being Element of NAT st
for n being Element of NAT st n >= m holds
||.((vseq . n) - tv).|| < e
consider m being Element of NAT such that
A67: for n being Element of NAT st n >= m holds
( |.((seq_id tseq) - (seq_id (vseq . n))).| (#) |.((seq_id tseq) - (seq_id (vseq . n))).| is summable & |.(Sum (((seq_id tseq) - (seq_id (vseq . n))) (#) (((seq_id tseq) - (seq_id (vseq . n))) *'))).| < e * e ) by A19, A66;

now

tseq in the_set_of_ComplexSequences by CSSPACE:def 1;
then reconsider u = tseq as VECTOR of Complex_l2_Space by A58, CSSPACE:def 11, CSSPACE:def 18;
let n be Element of NAT ; :: thesis: ( n >= m implies ||.((vseq . n) - tv).|| < e )
assume A68: n >= m ; :: thesis: ||.((vseq . n) - tv).|| < e
reconsider v = vseq . n as VECTOR of Complex_l2_Space ;
set seqvn = seq_id (vseq . n);
0 <= Re ((u - v) .|. (u - v)) by CSSPACE:def 13;
then A69: 0 <= |.((u - v) .|. (u - v)).| by CSSPACE:34;
seq_id (u - v) = u - v by Lm12;
then A70: seq_id (u - v) = (seq_id tseq) - (seq_id (vseq . n)) by Lm16;
|.(Sum (((seq_id tseq) - (seq_id (vseq . n))) (#) (((seq_id tseq) - (seq_id (vseq . n))) *'))).| < e * e by A67, A68;
then |.((u - v) .|. (u - v)).| < e * e by A70, Lm18;
then A71: sqrt |.((u - v) .|. (u - v)).| < sqrt (e ^2) by A69, SQUARE_1:27;
||.((vseq . n) - tv).|| = ||.(- (tv - (vseq . n))).|| by RLVECT_1:33
.= ||.(tv - (vseq . n)).|| by CSSPACE:47
.= sqrt |.((u - v) .|. (u - v)).| by CSSPACE:def 15 ;
hence ||.((vseq . n) - tv).|| < e by A66, A71, SQUARE_1:22; :: thesis: verum

end;

hence ex m being Element of NAT st
for n being Element of NAT st n >= m holds
||.((vseq . n) - tv).|| < e ; :: thesis: verum

end;

hence vseq is convergent by CLVECT_2:9; :: thesis: verum