deffunc H_{1}( 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 = H_{1}(n)
and

A3: for n being Nat holds s2 . n = H_{1}(n)
; :: thesis: s1 = s2

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 = H

A3: for n being Nat holds s2 . n = H

now :: thesis: for x being object st x in NAT holds

s1 . x = s2 . x

hence
s1 = s2
; :: thesis: verums1 . 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 = H_{1}(n)
by A2

.= s2 . x by A3 ; :: thesis: verum

end;assume x in NAT ; :: thesis: s1 . x = s2 . x

then reconsider n = x as Element of NAT ;

thus s1 . x = H

.= s2 . x by A3 ; :: thesis: verum