
theorem UApCl1:
  for T being naturally_generated non empty with_equivalence TopRelStr,
      A being Subset of T holds
    A is closed iff UAp A = A
  proof
    let T be naturally_generated non empty with_equivalence TopRelStr,
        A be Subset of T;
    thus A is closed implies UAp A = A
    proof
      assume A is closed; then
Z:    (LAp A`)` = A`` by OpIsLap;
      (LAp A`)` = (UAp A)`` by ROUGHS_1:28 .= UAp A;
      hence thesis by Z;
    end;
    assume
Z1: UAp A = A;
    (LAp A`)` = (UAp A)`` by ROUGHS_1:28 .= UAp A;
    hence thesis by Z1;
  end;
