theorem
  for f1,f2,Z st Z c= dom (f1+f2) & f1 is_differentiable_on Z & f2
  is_differentiable_on Z holds f1+f2 is_differentiable_on Z & for x st x in Z
  holds ((f1+f2)`|Z)/.x = diff(f1,x)+diff(f2,x)
proof
  let f1,f2,Z;
  assume that
A1: Z c= dom (f1+f2) and
A2: f1 is_differentiable_on Z & f2 is_differentiable_on Z;
  now
    let x0;
    assume x0 in Z;
    then f1 is_differentiable_in x0 & f2 is_differentiable_in x0 by A2,Th15;
    hence f1+f2 is_differentiable_in x0 by Th23;
  end;
  hence
A3: f1+f2 is_differentiable_on Z by A1,Th15;
  now
    let x;
    assume
A4: x in Z;
    then
A5: f1 is_differentiable_in x & f2 is_differentiable_in x by A2,Th15;
    thus ((f1+f2)`|Z)/.x = diff((f1+f2),x) by A3,A4,Def12
      .= diff(f1,x)+diff(f2,x) by A5,Th23;
  end;
  hence thesis;
end;
