reserve x,y,Y,Z for set,
  L for LATTICE,
  l for Element of L;

theorem Th19:
  for L being lower-bounded continuous LATTICE, X,Y being Subset
  of L st X is order-generating & X c= Y holds Y is order-generating
proof
  let L be lower-bounded continuous LATTICE, X,Y be Subset of L;
  assume that
A1: X is order-generating and
A2: X c= Y;
  let x be Element of L;
  thus ex_inf_of (uparrow x) /\ Y,L by YELLOW_0:17;
A3: ex_inf_of ((uparrow x) /\ Y),L by YELLOW_0:17;
  ex_inf_of (uparrow x),L by WAYBEL_0:39;
  then inf ((uparrow x) /\ Y) >= inf (uparrow x) by A3,XBOOLE_1:17,YELLOW_0:35;
  then
A4: inf ((uparrow x) /\ Y) >= x by WAYBEL_0:39;
  ex_inf_of ((uparrow x) /\ X),L by YELLOW_0:17;
  then inf ((uparrow x) /\ X) >= inf ((uparrow x) /\ Y) by A2,A3,XBOOLE_1:26
,YELLOW_0:35;
  then x >= inf ((uparrow x) /\ Y) by A1;
  hence thesis by A4,ORDERS_2:2;
end;
