let a, b, c, x0, x1, x2, x3, x4 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_different 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_different 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_different or [!f,x0,x1,x2,x3,x4!] = 0 )
assume x0,x1,x2,x3,x4 are_mutually_different ; :: thesis: [!f,x0,x1,x2,x3,x4!] = 0
then ( x0 <> x1 & x0 <> x2 & x0 <> x3 & x0 <> x4 & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 ) by ZFMISC_1:def 7;
then A2: ( x0,x1,x2,x3 are_mutually_different & x1,x2,x3,x4 are_mutually_different ) by 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, A2, Th29 ;
hence [!f,x0,x1,x2,x3,x4!] = 0 ; :: thesis: verum