let H1, H2 be Functional_Sequence of D,REAL; :: thesis: ( ( for n being Nat holds H1 . n = r (#) (H . n) ) & ( for n being Nat holds H2 . n = r (#) (H . n) ) implies H1 = H2 )
assume that
A1: for n being Nat holds H1 . n = r (#) (H . n) and
A2: for n being Nat holds H2 . n = r (#) (H . n) ; :: thesis: H1 = H2
now :: thesis: for n being Element of NAT holds H1 . n = H2 . n
let n be Element of NAT ; :: thesis: H1 . n = H2 . n
H1 . n = r (#) (H . n) by A1;
hence H1 . n = H2 . n by A2; :: thesis: verum
end;
hence H1 = H2 by FUNCT_2:63; :: thesis: verum