reserve Z for open Subset of REAL;

theorem Th7:
  for n be Nat, x be Real st x in Z holds (diff(exp_R,Z)
  .n).x = exp_R.x
proof
  let n be Nat;
  let x be Real;
  assume x in Z;
  then
A1: x in dom(exp_R | Z) by Th5;
  (diff(exp_R,Z).n).x=(exp_R | Z).x by Th6
    .=exp_R.x by A1,FUNCT_1:47;
  hence thesis;
end;
