
theorem
  for T being TopStruct, x, y being Point of T, X, Y being Subset of T
st X = {x} & Y = {y} holds (for V being Subset of T st V is open holds (x in V
  implies y in V)) implies Cl X c= Cl Y
proof
  let T be TopStruct, x, y be Point of T, X, Y be Subset of T;
  assume that
A1: X = {x} and
A2: Y = {y} & for V being Subset of T st V is open holds x in V implies y in V;
  let z be object;
  assume
A3: z in Cl X;
  for V being Subset of T st V is open holds z in V implies Y meets V
  proof
    let V be Subset of T;
    assume that
A4: V is open and
A5: z in V;
    X meets V by A3,A4,A5,PRE_TOPC:def 7;
    then x in V by A1,ZFMISC_1:50;
    hence thesis by A2,A4,ZFMISC_1:48;
  end;
  hence thesis by A3,PRE_TOPC:def 7;
end;
