
theorem Fourteen:
  for T being with_equivalence naturally_generated non empty TopRelStr,
      A being Subset of T holds
    Kurat14Set A = { A, UAp A, (UAp A)`, A`, (LAp A)`, LAp A }
  proof
    let T be with_equivalence naturally_generated non empty TopRelStr,
        A be Subset of T;
Z1: now let A be Subset of T;
      A-`-` = (LAp (A-))`` by ROUGHS_1:29; then
      A-`-` = UAp UAp A by ROUGHS_1:36
           .= UAp A;
      hence Kurat14Part A = { A, A-, A-`, A-`} \/ { A-, A-, A-` }
            by ENUMSET1:19
        .= { A, A-, A-`, A-`} \/ { A-, A-` } by ENUMSET1:30
        .= { A, A- } \/ { A-`, A-`} \/ { A-, A-` } by ENUMSET1:5
        .= { A, A- } \/ { A-` } \/ { A-, A-` } by ENUMSET1:29
        .= { A, A- } \/ ({ A-` } \/ { A-, A-` }) by XBOOLE_1:4
        .= { A, A- } \/ { A-`, A-, A-` } by ENUMSET1:2
        .= { A, A- } \/ { A-, A-`, A-` } by ENUMSET1:58
        .= { A, A- } \/ ({ A- } \/ { A-`, A-` }) by ENUMSET1:2
        .= { A, A- } \/ ({ A- } \/ { A-` }) by ENUMSET1:29
        .= { A, A- } \/ { A- } \/ { A-` } by XBOOLE_1:4
        .= { A, A-, A- } \/ { A-` } by ENUMSET1:3
        .= { A-, A-, A } \/ { A-` } by ENUMSET1:59
        .= { A, A- } \/ { A-` } by ENUMSET1:30
        .= { A, A-, A-` } by ENUMSET1:3;
    end; then
Z2: Kurat14Part A` = { A`, A`-, A`-` };
z3: Kurat14Part A = { A, A-, A-` } by Z1;
    A`-` = (LAp A)`` by ROUGHS_1:29;
    hence thesis by ENUMSET1:13,z3,Z2;
  end;
