let S1, S2 be Real_Sequence; :: thesis: ( ( for n being Element of NAT holds S1 . n = (H . n) . x ) & ( for n being Element of NAT holds S2 . n = (H . n) . x ) implies S1 = S2 )
assume A1: ( ( for n being Element of NAT holds S1 . n = (H . n) . x ) & ( for n being Element of NAT holds S2 . n = (H . n) . x ) ) ; :: thesis: S1 = S2

now

let n be Element of NAT ; :: thesis: S1 . n = S2 . n
( S1 . n = (H . n) . x & S2 . n = (H . n) . x ) by A1;
hence S1 . n = S2 . n ; :: thesis: verum

end;

hence S1 = S2 by FUNCT_2:113; :: thesis: verum