
theorem Th25:
  for T being non empty TopSpace st T is T_2 for S being sequence
  of T,x being Point of T holds S is_convergent_to x iff S is convergent & x =
  lim S
proof
  let T be non empty TopSpace;
  assume
A1: T is T_2;
  let S be sequence of T, x be Point of T;
  thus S is_convergent_to x implies S is convergent & x = lim S
  proof
    assume
A2: S is_convergent_to x;
    hence S is convergent;
    then ex y being Point of T st Lim S = {y} by A1,Th24;
    hence thesis by A2,Def2;
  end;
  assume that
A3: S is convergent and
A4: x = lim S;
  ex x being Point of T st Lim S = {x} by A1,A3,Th24;
  hence thesis by A4,Def2;
end;
