let h, a, b, c be Real; :: thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c ) holds

for x being Real holds (fD (f,h)) . x = ((((2 * a) * h) * x) + (a * (h ^2))) + (b * h)

let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c ) implies for x being Real holds (fD (f,h)) . x = ((((2 * a) * h) * x) + (a * (h ^2))) + (b * h) )

assume A1: for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c ; :: thesis: for x being Real holds (fD (f,h)) . x = ((((2 * a) * h) * x) + (a * (h ^2))) + (b * h)

let x be Real; :: thesis: (fD (f,h)) . x = ((((2 * a) * h) * x) + (a * (h ^2))) + (b * h)

(fD (f,h)) . x = (f . (x + h)) - (f . x) by DIFF_1:3

.= (((a * ((x + h) ^2)) + (b * (x + h))) + c) - (f . x) by A1

.= (((a * ((x + h) ^2)) + (b * (x + h))) + c) - (((a * (x ^2)) + (b * x)) + c) by A1

.= ((((2 * a) * h) * x) + (a * (h ^2))) + (b * h) ;

hence (fD (f,h)) . x = ((((2 * a) * h) * x) + (a * (h ^2))) + (b * h) ; :: thesis: verum

for x being Real holds (fD (f,h)) . x = ((((2 * a) * h) * x) + (a * (h ^2))) + (b * h)

let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c ) implies for x being Real holds (fD (f,h)) . x = ((((2 * a) * h) * x) + (a * (h ^2))) + (b * h) )

assume A1: for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c ; :: thesis: for x being Real holds (fD (f,h)) . x = ((((2 * a) * h) * x) + (a * (h ^2))) + (b * h)

let x be Real; :: thesis: (fD (f,h)) . x = ((((2 * a) * h) * x) + (a * (h ^2))) + (b * h)

(fD (f,h)) . x = (f . (x + h)) - (f . x) by DIFF_1:3

.= (((a * ((x + h) ^2)) + (b * (x + h))) + c) - (f . x) by A1

.= (((a * ((x + h) ^2)) + (b * (x + h))) + c) - (((a * (x ^2)) + (b * x)) + c) by A1

.= ((((2 * a) * h) * x) + (a * (h ^2))) + (b * h) ;

hence (fD (f,h)) . x = ((((2 * a) * h) * x) + (a * (h ^2))) + (b * h) ; :: thesis: verum