
theorem
  for T being with_properly_defined_Topology 2TopStruct,
      A being Subset of T holds
    A is op-open iff A is open
  proof
    let T be with_properly_defined_Topology 2TopStruct;
    let A be Subset of T;
    set f = the SecondOp of T;
    thus A is op-open implies A is open
    proof
      assume A is op-open; then
      A in the topology of T by PDTo;
      hence thesis by PRE_TOPC:def 2;
    end;
    assume A is open; then
    A in the topology of T by PRE_TOPC:def 2; then
    ex S being Subset of T st S = A & S is op-open by PDTo;
    hence thesis;
  end;
