
theorem Lem1:
  for T being with_properly_defined_topology 1TopStruct,
      A being Subset of T holds
    A is op-closed iff A is closed
  proof
    let T be with_properly_defined_topology 1TopStruct;
    let A be Subset of T;
    set f = the FirstOp of T;
    thus A is op-closed implies A is closed
    proof
      assume A is op-closed; then
      A` in the topology of T by PDT;
      hence thesis by TOPS_1:3,PRE_TOPC:def 2;
    end;
    thus A is closed implies A is op-closed
    proof
      assume A is closed; then
      A` in the topology of T by PRE_TOPC:def 2,TOPS_1:3; then
      consider S being Subset of T such that
K1:   S` = A` & S is op-closed by PDT;
      S`` = A`` by K1;
      hence thesis by K1;
    end;
  end;
