
theorem Th14: :: Theorem 1.3.2. (k)
  for T being TopSpace, A being Subset of T holds A is open closed
  iff Fr A = {}
proof
  let T be TopSpace, A be Subset of T;
  hereby
    assume
A1: A is open closed;
    then
A2: Int A = A by TOPS_1:23;
    Fr A = A \ Int A by A1,TOPS_1:43
      .= {} by A2,XBOOLE_1:37;
    hence Fr A = {};
  end;
  assume
A3: Fr A = {};
  Fr A = Cl A \ Int A by Th8;
  then
A4: Cl A c= Int A by A3,XBOOLE_1:37;
A5: Int A c= A by TOPS_1:16;
  then A c= Cl A & Cl A c= A by A4,PRE_TOPC:18;
  then Cl A = A;
  hence thesis by A4,A5,XBOOLE_0:def 10;
end;
