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 Th28:
  (for x holds f.x = a*x^2+b*x+c) & x0,x1,x2
  are_mutually_distinct implies [!f,x0,x1,x2!]=a
proof
  assume
A1: for x holds f.x = a*x^2+b*x+c;
  assume
A2: x0,x1,x2 are_mutually_distinct;
  then
A3: x1<>x2 by ZFMISC_1:def 5;
A4: x0-x2<>0 by A2,ZFMISC_1:def 5;
  x0<>x1 by A2,ZFMISC_1:def 5;
  then [!f,x0,x1,x2!] = (a*(x0+x1)+b-[!f,x1,x2!])/(x0-x2) by A1,Th27
    .= (a*(x0+x1)+b-(a*(x1+x2)+b))/(x0-x2) by A1,A3,Th27
    .= a*(x0-x2)/(x0-x2);
  hence thesis by A4,XCMPLX_1:89;
end;
