
theorem Th31:
  for m be non zero Element of NAT,
      f be PartFunc of REAL-NS m, REAL-NS m,
      x be Point of REAL-NS m
    st f is_differentiable_in x
  holds
    diff(f,x) is invertible
      iff
    Det(Jacobian(f,x)) <> 0.F_Real
proof
  let m be non zero Element of NAT,
      f be PartFunc of REAL-NS m, REAL-NS m,
      x be Point of REAL-NS m;
  assume f is_differentiable_in x;
  then
  diff(f,x) is invertible
    iff
  Jacobian(f,x) is invertible by Th30;

  hence thesis by LAPLACE:34;
end;
