reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;
reserve u,v for Element of Tarski-Class(X),
  A,B,C for Ordinal,
  L for Sequence;

theorem Th13:
  A <> {} & A is limit_ordinal implies
  (x in Tarski-Class(X,A) iff ex B st B in A & x in Tarski-Class(X,B))
proof
  assume
A1: A <> {} & A is limit_ordinal;
then A2: Tarski-Class(X,A) = { u : ex B st B in A & u in Tarski-Class(X,B) }
  by Th9;
  thus
  x in Tarski-Class(X,A) implies ex B st B in A & x in Tarski-Class(X,B)
  proof
    assume x in Tarski-Class(X,A);
then  ex u st x = u & ex B st B in A & u in Tarski-Class(X,B) by A2;
    hence thesis;
  end;
  given B such that
A3: B in A and
A4: x in Tarski-Class(X,B);
  reconsider u = x as Element of Tarski-Class X by A4;
 u in { v : ex B st B in A & v in Tarski-Class(X,B) } by A3,A4;
  hence thesis by A1,Th9;
end;
