
theorem UApCl:
  for T being naturally_generated non empty with_equivalence TopRelStr,
      A being Subset of T holds
    UAp A = Cl A
  proof
    let T be naturally_generated non empty with_equivalence TopRelStr,
        A be Subset of T;
    UAp A c= Cl A
    proof
      let x be object;
      assume
A1:   x in UAp A; then
      reconsider xx = x as set;
      for C being Subset of T st C is closed & A c= C holds xx in C
      proof
        let C be Subset of T;
        assume C is closed; then
B3:     UAp C = C by UApCl1;
        assume A c= C; then
        UAp A c= UAp C by ROUGHS_1:25;
        hence thesis by B3,A1;
      end;
      hence thesis by PRE_TOPC:15,A1;
    end;
    hence thesis by TOPS_1:5,ROUGHS_1:13;
  end;
