theorem Th13:
  (X is closed_wrt_A1-A7 & B is finite & for o st o in B holds o
  in X) implies B in X
proof
  defpred P[set] means $1 in X;
  assume that
A1: X is closed_wrt_A1-A7 and
A2: B is finite and
A3: for o st o in B holds o in X;
A4: B is finite by A2;
A5: for o,C being set st o in B & C c= B & P[C] holds P[C \/ {o}]
  proof
    let o,C be set;
    assume that
A6: o in B and
    C c= B and
A7: C in X;
    o in X by A3,A6;
    then {o} in X by A1,Th2;
    hence thesis by A1,A7,Th4;
  end;
A8: P[{}] by A1,Th3;
  thus P[B] from FINSET_1:sch 2(A4,A8,A5);
end;
