
theorem OpIsLap:
  for T being non empty TopRelStr st T is naturally_generated holds
    for A being Subset of T holds
      A is open iff LAp A = A
  proof
    let T be non empty TopRelStr;
    assume
A1: T is naturally_generated;
    let A be Subset of T;
    thus A is open implies LAp A = A
    proof
      assume A is open; then
      A in GenTop LAp T by A1,PRE_TOPC:def 2; then
      ex S being Subset of T st S = A & S is (LAp T)-closed by GTDef;
      hence thesis by ROUGHS_2:def 10;
    end; ::::::
    assume LAp A = A; then
    A is (LAp T)-closed by ROUGHS_2:def 10; then
    A in GenTop LAp T by GTDef;
    hence thesis by PRE_TOPC:def 2,A1;
  end;
