reserve y for set,
  x,a,b for Real,
  n for Element of NAT,
  Z for open Subset of REAL,
  f,f1,f2,g for PartFunc of REAL,REAL;

theorem Th19:
  Z c= dom ln implies ln is_differentiable_on Z & for x st x in Z
  holds (ln`|Z).x = 1/x
proof
  assume
A1: Z c= dom ln;
  then
A2: for x st x in Z holds ln is_differentiable_in x by Lm5,TAYLOR_1:18;
  then
A3: ln is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds (ln`|Z).x = 1/x
  proof
    let x;
    assume
A4: x in Z;
    then diff(ln,x) = 1/x by A1,TAYLOR_1:18;
    hence thesis by A3,A4,FDIFF_1:def 7;
  end;
  hence thesis by A1,A2,FDIFF_1:9;
end;
