
theorem
  for L being complete LATTICE st for x being Element of L holds x is compact
  holds L is satisfying_axiom_of_approximation
proof
  let L be complete LATTICE such that
A1: for x being Element of L holds x is compact;
  let x be Element of L;
  x is compact by A1;
  then x << x;
  then
A2: x in waybelow x;
  waybelow x is_<=_than sup waybelow x by YELLOW_0:32;
  then
A3: x <= sup waybelow x by A2;
A4: ex_sup_of waybelow x, L by YELLOW_0:17;
A5: ex_sup_of downarrow x, L by WAYBEL_0:34;
  sup downarrow x = x by WAYBEL_0:34;
  then x >= sup waybelow x by A4,A5,Th11,YELLOW_0:34;
  hence thesis by A3,ORDERS_2:2;
end;
