reserve Z for open Subset of REAL;

theorem
  for f be PartFunc of REAL,REAL, Z be Subset of REAL st f
  is_differentiable_on Z holds (-f) `| Z = -f `| Z
proof
  let f be PartFunc of REAL,REAL, Z be Subset of REAL such that
A1: f is_differentiable_on Z;
  Z is open Subset of REAL by A1,FDIFF_1:10;
  then (-f) `| Z = (-1)(#) (f `| Z) by A1,FDIFF_2:19
    .= -f `| Z;
  hence thesis;
end;
