let a be Real_Sequence; :: thesis: for n being Nat st ( for k being Nat st k <= n holds

a . k = 0 ) holds

(Partial_Sums a) . n = 0

defpred S_{1}[ Nat] means ( ( for k being Nat st k <= $1 holds

a . k = 0 ) implies (Partial_Sums a) . $1 = 0 );

A1: for n being Nat st S_{1}[n] holds

S_{1}[n + 1]
_{1}[ 0 ]
_{1}[n]
from NAT_1:sch 2(A6, A1); :: thesis: verum

a . k = 0 ) holds

(Partial_Sums a) . n = 0

defpred S

a . k = 0 ) implies (Partial_Sums a) . $1 = 0 );

A1: for n being Nat st S

S

proof

A6:
S
let n be Nat; :: thesis: ( S_{1}[n] implies S_{1}[n + 1] )

assume A2: S_{1}[n]
; :: thesis: S_{1}[n + 1]

_{1}[n + 1]
; :: thesis: verum

end;assume A2: S

now :: thesis: ( ( for k being Nat st k <= n + 1 holds

a . k = 0 ) implies (Partial_Sums a) . (n + 1) = 0 )

hence
Sa . k = 0 ) implies (Partial_Sums a) . (n + 1) = 0 )

assume A3:
for k being Nat st k <= n + 1 holds

a . k = 0 ; :: thesis: (Partial_Sums a) . (n + 1) = 0

.= 0 by A2, A3, A4 ; :: thesis: verum

end;a . k = 0 ; :: thesis: (Partial_Sums a) . (n + 1) = 0

A4: now :: thesis: for k being Nat st k <= n holds

a . k = 0

thus (Partial_Sums a) . (n + 1) =
((Partial_Sums a) . n) + (a . (n + 1))
by SERIES_1:def 1
a . k = 0

A5:
n <= n + 1
by NAT_1:11;

let k be Nat; :: thesis: ( k <= n implies a . k = 0 )

assume k <= n ; :: thesis: a . k = 0

hence a . k = 0 by A3, A5, XXREAL_0:2; :: thesis: verum

end;let k be Nat; :: thesis: ( k <= n implies a . k = 0 )

assume k <= n ; :: thesis: a . k = 0

hence a . k = 0 by A3, A5, XXREAL_0:2; :: thesis: verum

.= 0 by A2, A3, A4 ; :: thesis: verum

proof

thus
for n being Nat holds S
assume
for k being Nat st k <= 0 holds

a . k = 0 ; :: thesis: (Partial_Sums a) . 0 = 0

then a . 0 = 0 ;

hence (Partial_Sums a) . 0 = 0 by SERIES_1:def 1; :: thesis: verum

end;a . k = 0 ; :: thesis: (Partial_Sums a) . 0 = 0

then a . 0 = 0 ;

hence (Partial_Sums a) . 0 = 0 by SERIES_1:def 1; :: thesis: verum