reserve c for Complex;
reserve r for Real;
reserve m,n for Nat;
reserve f for complex-valued Function;

theorem Th9:
  for f being Function of REAL,REAL holds
  f is differentiable iff for r holds f is_differentiable_in r
  proof
    let f be Function of REAL,REAL;
A1: f|REAL = f;
    dom f = REAL by FUNCT_2:def 1;
    hence f is differentiable implies
    for r holds f is_differentiable_in r by A1,FDIFF_1:def 6,XREAL_0:def 1;
    assume for r holds f is_differentiable_in r;
    hence f is_differentiable_on dom f;
  end;
