let x0, x1, x2, x3, x4, 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,x4 are_mutually_distinct holds

[!f,x0,x1,x2,x3,x4!] = 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,x4 are_mutually_distinct implies [!f,x0,x1,x2,x3,x4!] = 0 )

assume A1: for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c ; :: thesis: ( not x0,x1,x2,x3,x4 are_mutually_distinct or [!f,x0,x1,x2,x3,x4!] = 0 )

assume A2: x0,x1,x2,x3,x4 are_mutually_distinct ; :: thesis: [!f,x0,x1,x2,x3,x4!] = 0

then A3: ( x1 <> x2 & x1 <> x3 ) by ZFMISC_1:def 7;

A4: x0 <> x3 by A2, ZFMISC_1:def 7;

A5: x2 <> x3 by A2, ZFMISC_1:def 7;

A6: x3 <> x4 by A2, ZFMISC_1:def 7;

( x1 <> x4 & x2 <> x4 ) by A2, ZFMISC_1:def 7;

then A7: x1,x2,x3,x4 are_mutually_distinct by A3, A5, A6, ZFMISC_1:def 6;

( x0 <> x1 & x0 <> x2 ) by A2, ZFMISC_1:def 7;

then x0,x1,x2,x3 are_mutually_distinct by A4, A3, A5, ZFMISC_1:def 6;

then [!f,x0,x1,x2,x3,x4!] = (0 - [!f,x1,x2,x3,x4!]) / (x0 - x4) by A1, Th29

.= (0 - 0) / (x0 - x4) by A1, A7, Th29 ;

hence [!f,x0,x1,x2,x3,x4!] = 0 ; :: thesis: verum

[!f,x0,x1,x2,x3,x4!] = 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,x4 are_mutually_distinct implies [!f,x0,x1,x2,x3,x4!] = 0 )

assume A1: for x being Real holds f . x = ((a * (x ^2)) + (b * x)) + c ; :: thesis: ( not x0,x1,x2,x3,x4 are_mutually_distinct or [!f,x0,x1,x2,x3,x4!] = 0 )

assume A2: x0,x1,x2,x3,x4 are_mutually_distinct ; :: thesis: [!f,x0,x1,x2,x3,x4!] = 0

then A3: ( x1 <> x2 & x1 <> x3 ) by ZFMISC_1:def 7;

A4: x0 <> x3 by A2, ZFMISC_1:def 7;

A5: x2 <> x3 by A2, ZFMISC_1:def 7;

A6: x3 <> x4 by A2, ZFMISC_1:def 7;

( x1 <> x4 & x2 <> x4 ) by A2, ZFMISC_1:def 7;

then A7: x1,x2,x3,x4 are_mutually_distinct by A3, A5, A6, ZFMISC_1:def 6;

( x0 <> x1 & x0 <> x2 ) by A2, ZFMISC_1:def 7;

then x0,x1,x2,x3 are_mutually_distinct by A4, A3, A5, ZFMISC_1:def 6;

then [!f,x0,x1,x2,x3,x4!] = (0 - [!f,x1,x2,x3,x4!]) / (x0 - x4) by A1, Th29

.= (0 - 0) / (x0 - x4) by A1, A7, Th29 ;

hence [!f,x0,x1,x2,x3,x4!] = 0 ; :: thesis: verum