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
  (for x 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
proof
  assume
A1: for x holds f.x = a*x^2+b*x+c;
  assume
A2: x0,x1,x2,x3,x4 are_mutually_distinct;
  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 qua Nat- [!f,x1,x2,x3,x4!])/(x0-x4) by A1,Th29
    .= (0 qua Nat-(0 qua Nat))/(x0-x4) by A1,A7,Th29;
  hence thesis;
end;
