let x0, x1, x2, x3, x4, a, b, c be Real; 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; ( ( 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
; ( 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
; [!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
; verum