let a, b, c, x0, x1, x2 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_different 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_different 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_different or [!f,x0,x1,x2!] = a )
assume A2: x0,x1,x2 are_mutually_different ; :: thesis: [!f,x0,x1,x2!] = a
then A3: ( x0 <> x1 & x0 <> x2 & x1 <> x2 ) by ZFMISC_1:def 5;
A4: ( x0 - x1 <> 0 & x0 - x2 <> 0 & x1 - x2 <> 0 ) by A2, ZFMISC_1:def 5;
[!f,x0,x1,x2!] = (((a * (x0 + x1)) + b) - [!f,x1,x2!]) / (x0 - x2) by A1, A3, 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_1:90; :: thesis: verum