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
  x0,x1,x2,x3 are_mutually_distinct implies [!AffineMap(a,b),x0,x1,x2, x3!]=0
proof
  assume
A1: x0,x1,x2,x3 are_mutually_distinct;
  then
A2: x1<>x2 by ZFMISC_1:def 6;
  x1<>x3 & x2<>x3 by A1,ZFMISC_1:def 6;
  then
A3: x1,x2,x3 are_mutually_distinct by A2,ZFMISC_1:def 5;
  x0<>x1 & x0<>x2 by A1,ZFMISC_1:def 6;
  then x0,x1,x2 are_mutually_distinct by A2,ZFMISC_1:def 5;
  then
  [!AffineMap(a,b),x0,x1,x2,x3!] = (0 qua Nat-[!AffineMap(a,b),x1,x2,x3!])
  /(x0-x3) by Th22
    .= (0 qua Nat-(0 qua Nat))/(x0-x3) by A3,Th22;
  hence thesis;
end;
