theorem
  f|X is_differentiable_on X iff f is_differentiable_on X
proof
  hereby assume
A1: f|X is_differentiable_on X; then
A2: for x being Real st x in X holds f|X is_differentiable_in x;
    X c= dom(f|X) by A1;
    then X c= dom f /\ X by RELAT_1:61;
    then X c= dom f by XBOOLE_1:18;
    hence f is_differentiable_on X by A2;
  end;
  thus thesis by RELAT_1:62;
end;
