reserve A, B, X, Y for set;

theorem
  for L1 being continuous antisymmetric non empty reflexive RelStr, L2
  being non empty reflexive RelStr st the RelStr of L1 = the RelStr of L2 holds
  L2 is continuous
proof
  let L1 be continuous antisymmetric non empty reflexive RelStr, L2 be non
  empty reflexive RelStr such that
A1: the RelStr of L1 = the RelStr of L2;
  hereby
    let y be Element of L2;
    reconsider x = y as Element of L1 by A1;
    waybelow x = waybelow y by A1,Th13;
    hence waybelow y is non empty directed by A1,WAYBEL_0:3;
  end;
  thus L2 is up-complete by A1,WAYBEL_8:15;
  let y be Element of L2;
  reconsider x = y as Element of L1 by A1;
A2: ex_sup_of waybelow x, L1 & x = sup waybelow x by WAYBEL_0:75,WAYBEL_3:def 5
;
  waybelow x = waybelow y by A1,Th13;
  hence thesis by A1,A2,YELLOW_0:26;
end;
