let S1, S2 be sequence of the carrier of Y; :: 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
A2: for n being Nat holds S1 . n = (H . n) /. x and
A3: 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
let n be Element of NAT ; :: thesis: S1 . n = S2 . n
S1 . n = (H . n) /. x by A2;
hence S1 . n = S2 . n by A3; :: thesis: verum
end;
hence S1 = S2 by FUNCT_2:63; :: thesis: verum