
theorem Th4:
  for L being non empty LattStr holds L is meet-Associative
  join-Associative meet-Absorbing join-absorbing implies L is meet-associative
  join-associative
proof
  let L be non empty LattStr;
  assume
A1: L is meet-Associative join-Associative meet-Absorbing join-absorbing;
  then
A2: L is meet-commutative join-commutative by Th2;
A3: for x,y,z being Element of L holds x "\/" (y "\/" z) = (x "\/" y) "\/" z
  proof
    let a,b,c be Element of L;
    a "\/" (b "\/" c) = a "\/" (c "\/" b) by A2
      .= c "\/" (a "\/" b) by A1
      .= (a "\/" b) "\/" c by A2;
    hence thesis;
  end;
  for x,y,z being Element of L holds x "/\" (y "/\" z) = (x "/\" y) "/\" z
  proof
    let a,b,c be Element of L;
    a "/\" (b "/\" c) = a "/\" (c "/\" b) by A2
      .= c "/\" (a "/\" b) by A1
      .= (a "/\" b) "/\" c by A2;
    hence thesis;
  end;
  hence thesis by A3;
end;
