let X be Complex_Banach_Algebra; :: thesis: for m being Nat

for seq1, seq2 being sequence of X st ( for n being Nat st n <= m holds

seq1 . n = seq2 . n ) holds

(Partial_Sums seq1) . m = (Partial_Sums seq2) . m

let m be Nat; :: thesis: for seq1, seq2 being sequence of X st ( for n being Nat st n <= m holds

seq1 . n = seq2 . n ) holds

(Partial_Sums seq1) . m = (Partial_Sums seq2) . m

let seq1, seq2 be sequence of X; :: thesis: ( ( for n being Nat st n <= m holds

seq1 . n = seq2 . n ) implies (Partial_Sums seq1) . m = (Partial_Sums seq2) . m )

defpred S_{1}[ Nat] means ( $1 <= m implies (Partial_Sums seq1) . $1 = (Partial_Sums seq2) . $1 );

assume A1: for n being Nat st n <= m holds

seq1 . n = seq2 . n ; :: thesis: (Partial_Sums seq1) . m = (Partial_Sums seq2) . m

A2: for k being Nat st S_{1}[k] holds

S_{1}[k + 1]
_{1}[ 0 ]
_{1}[k]
from NAT_1:sch 2(A5, A2);

hence (Partial_Sums seq1) . m = (Partial_Sums seq2) . m ; :: thesis: verum

for seq1, seq2 being sequence of X st ( for n being Nat st n <= m holds

seq1 . n = seq2 . n ) holds

(Partial_Sums seq1) . m = (Partial_Sums seq2) . m

let m be Nat; :: thesis: for seq1, seq2 being sequence of X st ( for n being Nat st n <= m holds

seq1 . n = seq2 . n ) holds

(Partial_Sums seq1) . m = (Partial_Sums seq2) . m

let seq1, seq2 be sequence of X; :: thesis: ( ( for n being Nat st n <= m holds

seq1 . n = seq2 . n ) implies (Partial_Sums seq1) . m = (Partial_Sums seq2) . m )

defpred S

assume A1: for n being Nat st n <= m holds

seq1 . n = seq2 . n ; :: thesis: (Partial_Sums seq1) . m = (Partial_Sums seq2) . m

A2: for k being Nat st S

S

proof

A5:
S
let k be Nat; :: thesis: ( S_{1}[k] implies S_{1}[k + 1] )

assume A3: S_{1}[k]
; :: thesis: S_{1}[k + 1]

assume A4: k + 1 <= m ; :: thesis: (Partial_Sums seq1) . (k + 1) = (Partial_Sums seq2) . (k + 1)

k < k + 1 by XREAL_1:29;

hence (Partial_Sums seq1) . (k + 1) = ((Partial_Sums seq2) . k) + (seq1 . (k + 1)) by A3, A4, BHSP_4:def 1, XXREAL_0:2

.= ((Partial_Sums seq2) . k) + (seq2 . (k + 1)) by A1, A4

.= (Partial_Sums seq2) . (k + 1) by BHSP_4:def 1 ;

:: thesis: verum

end;assume A3: S

assume A4: k + 1 <= m ; :: thesis: (Partial_Sums seq1) . (k + 1) = (Partial_Sums seq2) . (k + 1)

k < k + 1 by XREAL_1:29;

hence (Partial_Sums seq1) . (k + 1) = ((Partial_Sums seq2) . k) + (seq1 . (k + 1)) by A3, A4, BHSP_4:def 1, XXREAL_0:2

.= ((Partial_Sums seq2) . k) + (seq2 . (k + 1)) by A1, A4

.= (Partial_Sums seq2) . (k + 1) by BHSP_4:def 1 ;

:: thesis: verum

proof

for k being Nat holds S
assume
0 <= m
; :: thesis: (Partial_Sums seq1) . 0 = (Partial_Sums seq2) . 0

thus (Partial_Sums seq1) . 0 = seq1 . 0 by BHSP_4:def 1

.= seq2 . 0 by A1

.= (Partial_Sums seq2) . 0 by BHSP_4:def 1 ; :: thesis: verum

end;thus (Partial_Sums seq1) . 0 = seq1 . 0 by BHSP_4:def 1

.= seq2 . 0 by A1

.= (Partial_Sums seq2) . 0 by BHSP_4:def 1 ; :: thesis: verum

hence (Partial_Sums seq1) . m = (Partial_Sums seq2) . m ; :: thesis: verum