theorem Th15:
  Tarski-Class(X,A) c= Tarski-Class(X,succ A)
proof
  let x be object;
  assume x in Tarski-Class(X,A);
then  x in { u : ex v st v in Tarski-Class(X,A) & u c= v };
then A1: x in { u : ex v st v in Tarski-Class(X,A) & u c= v } \/
  { bool v : v in Tarski-Class(X,A) } by XBOOLE_0:def 3;
   Tarski-Class
(X,succ A) = { u : ex v st v in Tarski-Class(X,A) & u c= v } \/
  { bool v : v in Tarski-Class(X,A) } \/
  bool Tarski-Class(X,A) /\ Tarski-Class X by Lm1;
  hence thesis by A1,XBOOLE_0:def 3;
end;
