
theorem Th19:
  for K, L being non empty ComplLLattStr st the ComplLLattStr of K =
  the ComplLLattStr of L & K is with_Top holds L is with_Top
proof
  let K, L be non empty ComplLLattStr;
  assume that
A1: the ComplLLattStr of K = the ComplLLattStr of L and
A2: K is with_Top;
  for x, y being Element of L holds x |_| x` = y |_| y`
  proof
    let x, y be Element of L;
    reconsider x9 = x, y9 = y as Element of K by A1;
    x |_| x` = x9 |_| x9` by A1,Th18
      .= y9 |_| y9` by A2
      .= y |_| y` by A1,Th18;
    hence thesis;
  end;
  hence thesis;
end;
