
theorem
  for T being non empty TopSpace holds (for S being sequence of T holds
  for x1,x2 being Point of T holds (x1 in Lim S & x2 in Lim S implies x1=x2))
  implies T is T_1
proof
  let T be non empty TopSpace;
  assume
A1: for S being sequence of T holds for x1,x2 being Point of T holds (x1
  in Lim S & x2 in Lim S implies x1=x2);
  assume not T is T_1;
  then consider x1,x2 being Point of T,S being sequence of T such that
A2: S = (NAT --> x1) and
A3: x1 <> x2 and
A4: S is_convergent_to x2 by Lm3;
  S is_convergent_to x1 by A2,FRECHET:22;
  then
A5: x1 in Lim S by FRECHET:def 5;
  x2 in Lim S by A4,FRECHET:def 5;
  hence contradiction by A1,A3,A5;
end;
