reserve Z for open Subset of REAL;

theorem Th5:
  exp_R `| Z = exp_R | Z & dom(exp_R | Z) = Z
proof
A1: exp_R is_differentiable_on Z by FDIFF_1:26,TAYLOR_1:16;
A2: dom (exp_R | Z) = Z by Lm1;
A3: for x be Element of REAL st x in Z holds (exp_R `| Z).x=(exp_R | Z).x
  proof
    let x be Element of REAL such that
A4: x in Z;
    thus (exp_R `| Z).x=diff(exp_R,x) by A1,A4,FDIFF_1:def 7
      .=exp_R.x by TAYLOR_1:16
      .=(exp_R | Z).x by A2,A4,FUNCT_1:47;
  end;
  dom (exp_R `| Z) = Z by A1,FDIFF_1:def 7;
  hence thesis by A2,A3,PARTFUN1:5;
end;
