let x0, x1, x2, x3, 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,x3 are_mutually_distinct holds

[!f,x0,x1,x2,x3!] = 0

let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c ) & x0,x1,x2,x3 are_mutually_distinct implies [!f,x0,x1,x2,x3!] = 0 )

assume A1: for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c ; :: thesis: ( not x0,x1,x2,x3 are_mutually_distinct or [!f,x0,x1,x2,x3!] = 0 )

assume A2: x0,x1,x2,x3 are_mutually_distinct ; :: thesis: [!f,x0,x1,x2,x3!] = 0

then A3: x1 <> x2 by ZFMISC_1:def 6;

( x1 <> x3 & x2 <> x3 ) by A2, ZFMISC_1:def 6;

then A4: x1,x2,x3 are_mutually_distinct by A3, ZFMISC_1:def 5;

( x0 <> x1 & x0 <> x2 ) by A2, ZFMISC_1:def 6;

then x0,x1,x2 are_mutually_distinct by A3, ZFMISC_1:def 5;

then [!f,x0,x1,x2,x3!] = (a - [!f,x1,x2,x3!]) / (x0 - x3) by A1, Th28

.= (a - a) / (x0 - x3) by A1, A4, Th28 ;

hence [!f,x0,x1,x2,x3!] = 0 ; :: thesis: verum

[!f,x0,x1,x2,x3!] = 0

let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c ) & x0,x1,x2,x3 are_mutually_distinct implies [!f,x0,x1,x2,x3!] = 0 )

assume A1: for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c ; :: thesis: ( not x0,x1,x2,x3 are_mutually_distinct or [!f,x0,x1,x2,x3!] = 0 )

assume A2: x0,x1,x2,x3 are_mutually_distinct ; :: thesis: [!f,x0,x1,x2,x3!] = 0

then A3: x1 <> x2 by ZFMISC_1:def 6;

( x1 <> x3 & x2 <> x3 ) by A2, ZFMISC_1:def 6;

then A4: x1,x2,x3 are_mutually_distinct by A3, ZFMISC_1:def 5;

( x0 <> x1 & x0 <> x2 ) by A2, ZFMISC_1:def 6;

then x0,x1,x2 are_mutually_distinct by A3, ZFMISC_1:def 5;

then [!f,x0,x1,x2,x3!] = (a - [!f,x1,x2,x3!]) / (x0 - x3) by A1, Th28

.= (a - a) / (x0 - x3) by A1, A4, Th28 ;

hence [!f,x0,x1,x2,x3!] = 0 ; :: thesis: verum