
theorem Th30:
  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
    Jacobian(f,x) is invertible
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) = Mx2Tran(Jacobian(f,x)) by Th29;
  hence thesis by Th12;
end;
