
theorem Th15:
  for L1,L2 be non empty reflexive RelStr st the RelStr of L1 =
  the RelStr of L2 & L1 is up-complete holds L2 is up-complete
proof
  let L1,L2 be non empty reflexive RelStr;
  assume that
A1: the RelStr of L1 = the RelStr of L2 and
A2: L1 is up-complete;
  now
    let X be non empty directed Subset of L2;
    reconsider X9 = X as Subset of L1 by A1;
    reconsider X9 as non empty directed Subset of L1 by A1,WAYBEL_0:3;
    consider x9 be Element of L1 such that
A3: x9 is_>=_than X9 and
A4: for y9 be Element of L1 st y9 is_>=_than X9 holds x9 <= y9 by A2,
WAYBEL_0:def 39;
    reconsider x = x9 as Element of L2 by A1;
    take x;
    thus x is_>=_than X by A1,A3,YELLOW_0:2;
    let y be Element of L2 such that
A5: y is_>=_than X;
    reconsider y9 = y as Element of L1 by A1;
    x9 <= y9 by A1,A4,A5,YELLOW_0:2;
    hence x <= y by A1;
  end;
  hence thesis by WAYBEL_0:def 39;
end;
