reserve T for TopStruct;
reserve GX for TopSpace;

theorem Th22:
  for A being Subset of T holds (A is closed implies Cl A = A) & (
  T is TopSpace-like & Cl A = A implies A is closed)
proof
  let A be Subset of T;
  thus A is closed implies Cl A = A
  proof
    assume A is closed;
    then for x be object st x in Cl A holds x in A by Th15;
    then
A1: Cl A c= A;
    A c= Cl A by Th18;
    hence thesis by A1,XBOOLE_0:def 10;
  end;
  assume
A2: T is TopSpace-like;
  then consider F being Subset-Family of T such that
A3: for C being Subset of T holds C in F iff C is closed & A c= C and
A4: Cl A = meet F by Th16;
  assume
A5: Cl A = A;
  for C being Subset of T st C in F holds C is closed by A3;
  hence thesis by A2,A5,A4,Th14;
end;
