let x0, x1, 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 ) & x0 <> x1 holds

[!f,x0,x1!] = (a * (x0 + x1)) + b

let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c ) & x0 <> x1 implies [!f,x0,x1!] = (a * (x0 + x1)) + b )

assume that

A1: for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c and

A2: x0 <> x1 ; :: thesis: [!f,x0,x1!] = (a * (x0 + x1)) + b

A3: x0 - x1 <> 0 by A2;

[!f,x0,x1!] = ((((a * (x0 ^2)) + (b * x0)) + c) - (f . x1)) / (x0 - x1) by A1

.= ((((a * (x0 ^2)) + (b * x0)) + c) - (((a * (x1 ^2)) + (b * x1)) + c)) / (x0 - x1) by A1

.= (((a * (x0 + x1)) + b) * (x0 - x1)) / (x0 - x1) ;

hence [!f,x0,x1!] = (a * (x0 + x1)) + b by A3, XCMPLX_1:89; :: thesis: verum

[!f,x0,x1!] = (a * (x0 + x1)) + b

let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c ) & x0 <> x1 implies [!f,x0,x1!] = (a * (x0 + x1)) + b )

assume that

A1: for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c and

A2: x0 <> x1 ; :: thesis: [!f,x0,x1!] = (a * (x0 + x1)) + b

A3: x0 - x1 <> 0 by A2;

[!f,x0,x1!] = ((((a * (x0 ^2)) + (b * x0)) + c) - (f . x1)) / (x0 - x1) by A1

.= ((((a * (x0 ^2)) + (b * x0)) + c) - (((a * (x1 ^2)) + (b * x1)) + c)) / (x0 - x1) by A1

.= (((a * (x0 + x1)) + b) * (x0 - x1)) / (x0 - x1) ;

hence [!f,x0,x1!] = (a * (x0 + x1)) + b by A3, XCMPLX_1:89; :: thesis: verum