reserve f,f1,f2,g for PartFunc of REAL,REAL;
reserve A for non empty closed_interval Subset of REAL;
reserve p,r,x,x0 for Real;
reserve n for Element of NAT;
reserve Z for open Subset of REAL;

theorem Th23:
  for f1,Z st Z c= dom (-f1) & f1 is_differentiable_on Z holds -f1
  is_differentiable_on Z & for x st x in Z holds ((-f1)`|Z).x = -diff(f1,x)
proof
  let f1,Z;
  assume that
A1: Z c= dom (-f1) and
A2: f1 is_differentiable_on Z;
  now
    let x0;
    assume x0 in Z;
    then f1 is_differentiable_in x0 by A2,FDIFF_1:9;
    hence -f1 is_differentiable_in x0 by Th22;
  end;
  hence
A3: -f1 is_differentiable_on Z by A1,FDIFF_1:9;
  now
    let x;
    assume
A4: x in Z; then
A5: f1 is_differentiable_in x by A2,FDIFF_1:9;
    thus ((-f1)`|Z).x = diff((-f1),x) by A3,A4,FDIFF_1:def 7
      .= -diff(f1,x) by A5,Th22;
  end;
  hence thesis;
end;
