
theorem Th17:
  for L1,L2 be non empty reflexive antisymmetric RelStr st the
  RelStr of L1 = the RelStr of L2 & L1 is algebraic holds L2 is algebraic
proof
  let L1,L2 be non empty reflexive antisymmetric RelStr;
  assume that
A1: the RelStr of L1 = the RelStr of L2 and
A2: L1 is algebraic;
A3: L2 is up-complete by A1,A2,Th15;
A4: for x be Element of L2 holds compactbelow x is non empty directed
  proof
    let x be Element of L2;
    reconsider x9 = x as Element of L1 by A1;
    compactbelow x9 is non empty directed by A2;
    hence thesis by A1,A2,A3,Th10,WAYBEL_0:3;
  end;
  L2 is satisfying_axiom_K by A1,A2,A3,Th16;
  hence thesis by A3,A4;
end;
