theorem
  (for x be set st x in X holds f.x <> 0) & f is_differentiable_on X
  implies f^ is_differentiable_on X
proof
  assume that
A1: for x be set st x in X holds f.x <> 0 and
A2: f is_differentiable_on X;
  reconsider Z = X as Subset of REAL by A2,FDIFF_1:8;
  reconsider Z as open Subset of REAL by A2,FDIFF_1:10;
  for x be Real st x in Z holds f.x <> 0 by A1;
  hence thesis by A2,FDIFF_2:22;
end;
