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
  A <> {} & A is limit_ordinal & Y in Tarski-Class(X,A) &
  (Z c= Y or Z = bool Y) implies Z in Tarski-Class(X,A)
proof
  assume that
A1: A <> {} and
A2: A is limit_ordinal and
A3: Y in Tarski-Class(X,A);
  consider B such that
A4: B in A and
A5: Y in Tarski-Class(X,B) by A1,A2,A3,Th13;
A6: bool Y in Tarski-Class(X,succ B) by A5,Th10;
A7: Z c= Y implies Z in Tarski-Class(X,succ B) by A5,Th10;
A8: succ B in A by A2,A4,ORDINAL1:28;
  assume Z c= Y or Z = bool Y;
  hence thesis by A2,A6,A7,A8,Th13;
end;
