
theorem
  for L1,L2 being up-complete antisymmetric non empty reflexive RelStr
  st the RelStr of L1 = the RelStr of L2 & for x being Element of L1 holds
waybelow x is directed non empty holds L1 is satisfying_axiom_of_approximation
  implies L2 is satisfying_axiom_of_approximation
proof
  let L1,L2 be up-complete antisymmetric non empty reflexive RelStr such
  that
A1: the RelStr of L1 = the RelStr of L2 and
A2: for x being Element of L1 holds waybelow x is directed non empty and
A3: for x being Element of L1 holds x = sup waybelow x;
  let x be Element of L2;
  reconsider y = x as Element of L1 by A1;
A4: y = sup waybelow y by A3;
  waybelow y is directed non empty by A2;
  then
A5: ex_sup_of waybelow y, L1 by WAYBEL_0:75;
  waybelow y = waybelow x by A1,YELLOW12:13;
  hence thesis by A4,A5,A1,YELLOW_0:26;
end;
