deffunc H₁( Nat) -> set = ((((diff (f,Z)) . $1) . a) * ((b - a) |^ $1)) / ($1 !);
let s1, s2 be Real_Sequence; :: thesis: ( ( for n being Nat holds s1 . n = ((((diff (f,Z)) . n) . a) * ((b - a) |^ n)) / (n !) ) & ( for n being Nat holds s2 . n = ((((diff (f,Z)) . n) . a) * ((b - a) |^ n)) / (n !) ) implies s1 = s2 )
assume that
A2: for n being Nat holds s1 . n = H₁(n) and
A3: for n being Nat holds s2 . n = H₁(n) ; :: thesis: s1 = s2

now :: thesis: for x being object st x in NAT holds
s1 . x = s2 . x

let x be object ; :: thesis: ( x in NAT implies s1 . x = s2 . x )
assume x in NAT ; :: thesis: s1 . x = s2 . x
then reconsider n = x as Element of NAT ;
thus s1 . x = H₁(n) by A2
.= s2 . x by A3 ; :: thesis: verum

end;

hence s1 = s2 ; :: thesis: verum