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

proof

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;

hence S1 = S2 ; :: thesis: verum