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

now

let x be set ; :: thesis: ( x in NAT implies s1 . x = s2 . x )
assume A4: x in NAT ; :: thesis: s1 . x = s2 . x
reconsider n = x as Element of NAT by A4;
thus s1 . x = H₁(n) by A2
.= s2 . x by A3 ; :: thesis: verum

end;

hence s1 = s2 by FUNCT_2:18; :: thesis: verum