let x0, x1, x2, 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,x2 are_mutually_distinct holds

[!f,x0,x1,x2!] = a

let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c ) & x0,x1,x2 are_mutually_distinct implies [!f,x0,x1,x2!] = a )

assume A1: for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c ; :: thesis: ( not x0,x1,x2 are_mutually_distinct or [!f,x0,x1,x2!] = a )

assume A2: x0,x1,x2 are_mutually_distinct ; :: thesis: [!f,x0,x1,x2!] = a

then A3: x1 <> x2 by ZFMISC_1:def 5;

A4: x0 - x2 <> 0 by A2, ZFMISC_1:def 5;

x0 <> x1 by A2, ZFMISC_1:def 5;

then [!f,x0,x1,x2!] = (((a * (x0 + x1)) + b) - [!f,x1,x2!]) / (x0 - x2) by A1, Th27

.= (((a * (x0 + x1)) + b) - ((a * (x1 + x2)) + b)) / (x0 - x2) by A1, A3, Th27

.= (a * (x0 - x2)) / (x0 - x2) ;

hence [!f,x0,x1,x2!] = a by A4, XCMPLX_1:89; :: thesis: verum

[!f,x0,x1,x2!] = a

let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c ) & x0,x1,x2 are_mutually_distinct implies [!f,x0,x1,x2!] = a )

assume A1: for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c ; :: thesis: ( not x0,x1,x2 are_mutually_distinct or [!f,x0,x1,x2!] = a )

assume A2: x0,x1,x2 are_mutually_distinct ; :: thesis: [!f,x0,x1,x2!] = a

then A3: x1 <> x2 by ZFMISC_1:def 5;

A4: x0 - x2 <> 0 by A2, ZFMISC_1:def 5;

x0 <> x1 by A2, ZFMISC_1:def 5;

then [!f,x0,x1,x2!] = (((a * (x0 + x1)) + b) - [!f,x1,x2!]) / (x0 - x2) by A1, Th27

.= (((a * (x0 + x1)) + b) - ((a * (x1 + x2)) + b)) / (x0 - x2) by A1, A3, Th27

.= (a * (x0 - x2)) / (x0 - x2) ;

hence [!f,x0,x1,x2!] = a by A4, XCMPLX_1:89; :: thesis: verum