let vseq be sequence of Complex_l1_Space; :: thesis:
assume A1: vseq is Cauchy_sequence_by_Norm ; :: thesis:
defpred S₁[ object , object ] means ex i being Nat st
( $1 = i & ex rseqi being Complex_Sequence st
( ( for n being Nat holds rseqi . n = (seq_id (vseq . n)) . i ) & rseqi is convergent & $2 = lim rseqi ) );
A2: for x being object st x in NAT holds
ex y being object st
( y in COMPLEX & S₁[x,y] )

proof

let x be object ; :: thesis:
assume x in NAT ; :: thesis:
then reconsider i = x as Element of NAT ;
deffunc H₁( Nat) -> Element of COMPLEX = (seq_id (vseq . $1)) . i;
consider rseqi being Complex_Sequence such that
A3: for n being Nat holds rseqi . n = H₁(n) from COMSEQ_1:sch 1();
take lim rseqi ; :: thesis:
thus lim rseqi in COMPLEX by XCMPLX_0:def 2; :: thesis:

now :: thesis:

let e be Real; :: thesis:
assume A4: e > 0 ; :: thesis:
thus ex k being Nat st
for m being Nat st k <= m holds
|.((rseqi . m) - (rseqi . k)).| < e :: thesis:

proof

consider k being Nat such that
A5: for n, m being Nat st n >= k & m >= k holds
||.((vseq . n) - (vseq . m)).|| < e by A1, A4, Th7;
for m being Nat st k <= m holds
|.((rseqi . m) - (rseqi . k)).| < e

proof

let m be Nat; :: thesis:
assume A6: k <= m ; :: thesis:
A7: for i being Nat holds 0 <= |.(seq_id ((vseq . m) - (vseq . k))).| . i

proof

let i be Nat; :: thesis:
0 <= |.((seq_id ((vseq . m) - (vseq . k))) . i).| by COMPLEX1:46;
hence 0 <= |.(seq_id ((vseq . m) - (vseq . k))).| . i by VALUED_1:18; :: thesis:

end;

seq_id ((vseq . m) - (vseq . k)) is absolutely_summable by Def1;
then |.(seq_id ((vseq . m) - (vseq . k))).| is summable by COMSEQ_3:def 9;
then A8: |.(seq_id ((vseq . m) - (vseq . k))).| . i <= Sum |.(seq_id ((vseq . m) - (vseq . k))).| by A7, RSSPACE2:3;
seq_id ((vseq . m) - (vseq . k)) = seq_id ((seq_id (vseq . m)) - (seq_id (vseq . k))) by Th5
.= (seq_id (vseq . m)) + (- (seq_id (vseq . k))) ;
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)
.= (rseqi . m) - ((seq_id (vseq . k)) . i) by A3
.= (rseqi . m) - (rseqi . k) by A3 ;
then A9: |.((rseqi . m) - (rseqi . k)).| = |.(seq_id ((vseq . m) - (vseq . k))).| . i by VALUED_1:18;
||.((vseq . m) - (vseq . k)).|| = Sum |.(seq_id ((vseq . m) - (vseq . k))).| by Th5;
then Sum |.(seq_id ((vseq . m) - (vseq . k))).| < e by A5, A6;
hence |.((rseqi . m) - (rseqi . k)).| < e by A8, A9, XXREAL_0:2; :: thesis:

end;

hence ex k being Nat st
for m being Nat st k <= m holds
|.((rseqi . m) - (rseqi . k)).| < e ; :: thesis:

end;

then rseqi is convergent by COMSEQ_3:46;
hence S₁[x, lim rseqi] by A3; :: thesis:

end;

consider f being sequence of COMPLEX such that
A10: for x being object st x in NAT holds
S₁[x,f . x] from FUNCT_2:sch 1(A2);
reconsider tseq = f as Complex_Sequence ;

A11: now :: thesis:

let i be Nat; :: thesis:
reconsider x = i as set ;
i in NAT by ORDINAL1:def 12;
then ex i0 being Nat st
( x = i0 & ex rseqi being Complex_Sequence st
( ( for n being Nat holds rseqi . n = (seq_id (vseq . n)) . i0 ) & rseqi is convergent & f . x = lim rseqi ) ) by A10;
hence ex rseqi being Complex_Sequence st
( ( for n being Nat holds rseqi . n = (seq_id (vseq . n)) . i ) & rseqi is convergent & tseq . i = lim rseqi ) ; :: thesis:

end;

A12: for e being Real st e > 0 holds
ex k being Nat st
for n being Nat st n >= k holds
( |.((seq_id tseq) - (seq_id (vseq . n))).| is summable & Sum |.((seq_id tseq) - (seq_id (vseq . n))).| < e )

proof

let e1 be Real; :: thesis:
assume A13: e1 > 0 ; :: thesis:
set e = e1 / 2;
A14: e1 / 2 < e1 by A13, XREAL_1:216;
e1 / 2 > 0 by A13, XREAL_1:215;
then consider k being Nat such that
A15: for n, m being Nat st n >= k & m >= k holds
||.((vseq . n) - (vseq . m)).|| < e1 / 2 by A1, Th7;
A16: for m, n being Nat st n >= k & m >= k holds
( |.(seq_id ((vseq . n) - (vseq . m))).| is summable & Sum |.(seq_id ((vseq . n) - (vseq . m))).| < e1 / 2 & ( for i being Nat holds 0 <= |.(seq_id ((vseq . n) - (vseq . m))).| . i ) )

proof

let m, n be Nat; :: thesis:
assume ( n >= k & m >= k ) ; :: thesis:
then ||.((vseq . n) - (vseq . m)).|| < e1 / 2 by A15;
then A17: the normF of Complex_l1_Space . ((vseq . n) - (vseq . m)) < e1 / 2 ;
A18: for i being Nat holds 0 <= |.(seq_id ((vseq . n) - (vseq . m))).| . i

proof

let i be Nat; :: thesis:
0 <= |.((seq_id ((vseq . n) - (vseq . m))) . i).| by COMPLEX1:46;
hence 0 <= |.(seq_id ((vseq . n) - (vseq . m))).| . i by VALUED_1:18; :: thesis:

end;

seq_id ((vseq . n) - (vseq . m)) is absolutely_summable by Def1;
hence ( |.(seq_id ((vseq . n) - (vseq . m))).| is summable & Sum |.(seq_id ((vseq . n) - (vseq . m))).| < e1 / 2 & ( for i being Nat holds 0 <= |.(seq_id ((vseq . n) - (vseq . m))).| . i ) ) by A17, A18, Def2, COMSEQ_3:def 9; :: thesis:

end;

A19: for n, i being Nat
for rseq being Real_Sequence st ( for m being Nat holds rseq . m = (Partial_Sums |.(seq_id ((vseq . m) - (vseq . n))).|) . i ) holds
( rseq is convergent & lim rseq = (Partial_Sums |.((seq_id tseq) - (seq_id (vseq . n))).|) . i )

proof

let n be Nat; :: thesis:
defpred S₂[ Nat] means for rseq being Real_Sequence st ( for m being Nat holds rseq . m = (Partial_Sums |.(seq_id ((vseq . m) - (vseq . n))).|) . $1 ) holds
( rseq is convergent & lim rseq = (Partial_Sums |.((seq_id tseq) - (seq_id (vseq . n))).|) . $1 );
A20: for m, k being Nat holds seq_id ((vseq . m) - (vseq . k)) = (seq_id (vseq . m)) - (seq_id (vseq . k))

proof

let m, k be Nat; :: thesis:
seq_id ((vseq . m) - (vseq . k)) = seq_id ((seq_id (vseq . m)) - (seq_id (vseq . k))) by Th5;
hence seq_id ((vseq . m) - (vseq . k)) = (seq_id (vseq . m)) - (seq_id (vseq . k)) ; :: thesis:

end;

now :: thesis:

let i be Nat; :: thesis:
assume A21: for rseq being Real_Sequence st ( for m being Nat holds rseq . m = (Partial_Sums |.(seq_id ((vseq . m) - (vseq . n))).|) . i ) holds
( rseq is convergent & lim rseq = (Partial_Sums |.((seq_id tseq) - (seq_id (vseq . n))).|) . i ) ; :: thesis:
thus for rseq being Real_Sequence st ( for m being Nat holds rseq . m = (Partial_Sums |.(seq_id ((vseq . m) - (vseq . n))).|) . (i + 1) ) holds
( rseq is convergent & lim rseq = (Partial_Sums |.((seq_id tseq) - (seq_id (vseq . n))).|) . (i + 1) ) :: thesis:

proof

deffunc H₁( Nat) -> set = (Partial_Sums |.(seq_id ((vseq . $1) - (vseq . n))).|) . i;
consider rseqb being Real_Sequence such that
A22: for m being Nat holds rseqb . m = H₁(m) from SEQ_1:sch 1();
consider rseq0 being Complex_Sequence such that
A23: for m being Nat holds rseq0 . m = (seq_id (vseq . m)) . (i + 1) and
A24: rseq0 is convergent and
A25: tseq . (i + 1) = lim rseq0 by A11;
let rseq be Real_Sequence; :: thesis:
assume A26: for m being Nat holds rseq . m = (Partial_Sums |.(seq_id ((vseq . m) - (vseq . n))).|) . (i + 1) ; :: thesis:

A27: now :: thesis:

end;

A28: rseqb is convergent by A21, A22;
then lim rseq = |.((lim rseq0) - ((seq_id (vseq . n)) . (i + 1))).| + (lim rseqb) by A24, A27, Lm3
.= |.((tseq . (i + 1)) + (- ((seq_id (vseq . n)) . (i + 1)))).| + (lim rseqb) by A25
.= |.((tseq . (i + 1)) + ((- (seq_id (vseq . n))) . (i + 1))).| + (lim rseqb) by VALUED_1:8
.= |.((tseq - (seq_id (vseq . n))) . (i + 1)).| + (lim rseqb) by VALUED_1:1
.= (|.(tseq - (seq_id (vseq . n))).| . (i + 1)) + (lim rseqb) by VALUED_1:18
.= (|.(tseq - (seq_id (vseq . n))).| . (i + 1)) + ((Partial_Sums |.((seq_id tseq) - (seq_id (vseq . n))).|) . i) by A21, A22
.= (Partial_Sums |.((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))).|) . (i + 1) ) by A28, A24, A27, Lm3; :: thesis:

end;

then A29: for i being Nat st S₂[i] holds
S₂[i + 1] ;

now :: thesis:

let rseq be Real_Sequence; :: thesis:
assume A30: for m being Nat holds rseq . m = (Partial_Sums |.(seq_id ((vseq . m) - (vseq . n))).|) . 0 ; :: thesis:
thus ( rseq is convergent & lim rseq = (Partial_Sums |.((seq_id tseq) - (seq_id (vseq . n))).|) . 0 ) :: thesis:

proof

consider rseq0 being Complex_Sequence such that
A31: for m being Nat holds rseq0 . m = (seq_id (vseq . m)) . 0 and
A32: rseq0 is convergent and
A33: tseq . 0 = lim rseq0 by A11;
A34: for m being Nat holds rseq . m = |.((rseq0 . m) - ((seq_id (vseq . n)) . 0)).|

proof

end;

end;

then A35: S₂[ 0 ] ;
for i being Nat holds S₂[i] from NAT_1:sch 2(A35, A29);
hence for i being Nat
for rseq being Real_Sequence st ( for m being Nat holds rseq . m = (Partial_Sums |.(seq_id ((vseq . m) - (vseq . n))).|) . i ) holds
( rseq is convergent & lim rseq = (Partial_Sums |.((seq_id tseq) - (seq_id (vseq . n))).|) . i ) ; :: thesis:

end;

for n being Nat st n >= k holds
( |.((seq_id tseq) - (seq_id (vseq . n))).| is summable & Sum |.((seq_id tseq) - (seq_id (vseq . n))).| < e1 )

proof

let n be Nat; :: thesis:
assume A36: n >= k ; :: thesis:
A37: for i being Nat st 0 <= i holds
(Partial_Sums |.((seq_id tseq) - (seq_id (vseq . n))).|) . i <= e1 / 2

proof

let i be Nat; :: thesis:
assume 0 <= i ; :: thesis:
deffunc H₁( Nat) -> set = (Partial_Sums |.(seq_id ((vseq . $1) - (vseq . n))).|) . i;
consider rseq being Real_Sequence such that
A38: for m being Nat holds rseq . m = H₁(m) from SEQ_1:sch 1();
A39: for m being Nat st m >= k holds
rseq . m <= e1 / 2

proof

end;

( rseq is convergent & lim rseq = (Partial_Sums |.((seq_id tseq) - (seq_id (vseq . n))).|) . i ) by A19, A38;
hence (Partial_Sums |.((seq_id tseq) - (seq_id (vseq . n))).|) . i <= e1 / 2 by A39, RSSPACE2:5; :: thesis:

end;

now :: thesis:

take e1 = e1; :: thesis:
let i be Nat; :: thesis:
(Partial_Sums |.((seq_id tseq) - (seq_id (vseq . n))).|) . i <= e1 / 2 by A37, NAT_1:2;
hence (Partial_Sums |.((seq_id tseq) - (seq_id (vseq . n))).|) . i < e1 by A14, XXREAL_0:2; :: thesis:

end;

then A43: Partial_Sums |.((seq_id tseq) - (seq_id (vseq . n))).| is bounded_above by SEQ_2:def 3;
A44: for i being Nat holds 0 <= |.((seq_id tseq) - (seq_id (vseq . n))).| . i

proof

end;

end;

hence ex k being Nat st
for n being Nat st n >= k holds
( |.((seq_id tseq) - (seq_id (vseq . n))).| is summable & Sum |.((seq_id tseq) - (seq_id (vseq . n))).| < e1 ) ; :: thesis:

end;

|.(seq_id tseq).| is summable

proof

set d = |.(seq_id tseq).|;
A45: for i being Nat holds 0 <= |.(seq_id tseq).| . i

proof

end;

proof

end;

end;

then A51: seq_id tseq is absolutely_summable by COMSEQ_3:def 9;
A52: tseq in the_set_of_ComplexSequences by FUNCT_2:8;
then reconsider tv = tseq as Point of Complex_l1_Space by A51, Def1;
for e being Real st e > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((vseq . n) - tv).|| < e

proof

let e be Real; :: thesis:
assume e > 0 ; :: thesis:
then consider m being Nat such that
A53: for n being Nat st n >= m holds
( |.((seq_id tseq) - (seq_id (vseq . n))).| is summable & Sum |.((seq_id tseq) - (seq_id (vseq . n))).| < e ) by A12;

now :: thesis:

reconsider u = tseq as VECTOR of Complex_l1_Space by A51, A52, Def1;
let n be Nat; :: thesis:
assume n >= m ; :: thesis:
then A54: Sum |.((seq_id tseq) - (seq_id (vseq . n))).| < e by A53;
reconsider v = vseq . n as VECTOR of Complex_l1_Space ;
seq_id (u - v) = u - v by Th5;
then Sum |.(seq_id (u - v)).| = Sum |.((seq_id tseq) - (seq_id (vseq . n))).| by Th5;
then A55: the normF of Complex_l1_Space . (u - v) < e by A54, Def2;
||.((vseq . n) - tv).|| = ||.(- (tv - (vseq . n))).|| by RLVECT_1:33
.= ||.(tv - (vseq . n)).|| by CLVECT_1:103 ;
hence ||.((vseq . n) - tv).|| < e by A55; :: thesis:

end;

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

end;

hence vseq is convergent by CLVECT_1:def 15; :: thesis: