theorem
  X c= Y implies Lim X c= Lim Y
proof
  assume
A1: X c= Y;
  let x be object;
  assume x in Lim X;
  then x in X & ex A st x = A & A is limit_ordinal by ORDINAL1:def 10;
  hence thesis by A1,ORDINAL1:def 10;
end;
