reserve V for Universe,
  a,b,x,y,z,x9,y9 for Element of V,
  X for Subclass of V,
  o,p,q,r,s,t,u,a1,a2,a3,A,B,C,D for set,
  K,L,M for Ordinal,
  n for Element of omega,
  fs for finite Subset of omega,
  e,g,h for Function,
  E for non empty set,
  f for Function of VAR,E,
  k,k1 for Element of NAT,
  v1,v2,v3 for Element of VAR,
  H,H9 for ZF-formula;

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;
