let S1, S2 be Real_Sequence; :: thesis: ( ( for n being Nat holds S1 . n = (S . n) - x ) & ( for n being Nat holds S2 . n = (S . n) - x ) implies S1 = S2 )

assume that

A2: for n being Nat holds S1 . n = (S . n) - x and

A3: for n being Nat holds S2 . n = (S . n) - x ; :: thesis: S1 = S2

for n being Nat holds S1 . n = S2 . n

assume that

A2: for n being Nat holds S1 . n = (S . n) - x and

A3: for n being Nat holds S2 . n = (S . n) - x ; :: thesis: S1 = S2

for n being Nat holds S1 . n = S2 . n

proof

hence
S1 = S2
; :: thesis: verum
let n be Nat; :: thesis: S1 . n = S2 . n

S1 . n = (S . n) - x by A2;

hence S1 . n = S2 . n by A3; :: thesis: verum

end;S1 . n = (S . n) - x by A2;

hence S1 . n = S2 . n by A3; :: thesis: verum