reserve A, B, X, Y for set;

theorem
  for N being transitive RelStr, A, J being Subset of N st A
  is_coarser_than uparrow J holds uparrow A c= uparrow J
proof
  let N be transitive RelStr, A, J be Subset of N such that
A1: A is_coarser_than uparrow J;
  let w be object;
  assume
A2: w in uparrow A;
  then reconsider w1 = w as Element of N;
  consider t being Element of N such that
A3: t <= w1 and
A4: t in A by A2,WAYBEL_0:def 16;
  consider t1 being Element of N such that
A5: t1 in uparrow J and
A6: t1 <= t by A1,A4,YELLOW_4:def 2;
  consider t2 being Element of N such that
A7: t2 <= t1 and
A8: t2 in J by A5,WAYBEL_0:def 16;
  t2 <= t by A6,A7,ORDERS_2:3;
  then t2 <= w1 by A3,ORDERS_2:3;
  hence thesis by A8,WAYBEL_0:def 16;
end;
