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

assume that

A3: for n being Nat holds S1 . n = (H . n) . x and

A4: for n being Nat holds S2 . n = (H . n) . x ; :: thesis: S1 = S2

assume that

A3: for n being Nat holds S1 . n = (H . n) . x and

A4: for n being Nat holds S2 . n = (H . n) . x ; :: thesis: S1 = S2

now :: thesis: for n being Element of NAT holds S1 . n = S2 . n

hence
S1 = S2
by FUNCT_2:63; :: thesis: verumlet n be Element of NAT ; :: thesis: S1 . n = S2 . n

S1 . n = (H . n) . x by A3;

hence S1 . n = S2 . n by A4; :: thesis: verum

end;S1 . n = (H . n) . x by A3;

hence S1 . n = S2 . n by A4; :: thesis: verum