reserve h,r,r1,r2,x0,x1,x2,x3,x4,x5,x,a,b,c,k for Real,
  f,f1,f2 for Function of REAL,REAL;

theorem Th29:
  (for x holds f.x = a*x^2+b*x+c) & x0,x1,x2,x3
  are_mutually_distinct implies [!f,x0,x1,x2,x3!]=0
proof
  assume
A1: for x holds f.x = a*x^2+b*x+c;
  assume
A2: x0,x1,x2,x3 are_mutually_distinct;
  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 thesis;
end;
