reserve X for TopSpace;
reserve C for Subset of X;
reserve A, B for Subset of X;

theorem Th5:
  A is open closed iff Cl A = Int A
proof
  thus A is open & A is closed implies Cl A = Int A
  proof
    assume that
A1: A is open and
A2: A is closed;
    Int A = A by A1,TOPS_1:23;
    hence thesis by A2,PRE_TOPC:22;
  end;
A3: Int A c= A by TOPS_1:16;
A4: A c= Cl A by PRE_TOPC:18;
  assume Cl A = Int A;
  hence thesis by A3,A4,XBOOLE_0:def 10;
end;
