
theorem Th2:
  for L being non empty LattStr holds L is meet-Associative
  join-Associative meet-Absorbing join-absorbing implies L is meet-commutative
  join-commutative
proof
  let L be non empty LattStr;
  assume
A1: L is meet-Associative join-Associative meet-Absorbing join-absorbing;
  then
A2: L is join-idempotent meet-idempotent by Th1;
A3: for x,y being Element of L holds x "/\" y = y "/\" x
  proof
    let a,b be Element of L;
    a "/\" b = a "/\" (b "/\" (b "\/" a)) by A1
      .= b "/\" (a "/\" (b "\/" a)) by A1
      .= b "/\" (a "/\" (b "\/" (a "\/" a))) by A2,ROBBINS1:def 7
      .= b "/\" (a "/\" (a "\/" (b "\/" a))) by A1
      .= b "/\" a by A1;
    hence thesis;
  end;
  for x,y being Element of L holds x "\/" y = y "\/" x
  proof
    let a,b be Element of L;
    a "\/" b = a "\/" (b "\/" (b "/\" a)) by A1
      .= b "\/" (a "\/" (b "/\" a)) by A1
      .= b "\/" (a "\/" (b "/\" (a "/\" a))) by A2,SHEFFER1:def 9
      .= b "\/" (a "\/" (a "/\" (b "/\" a))) by A1
      .= b "\/" a by A1;
    hence thesis;
  end;
  hence thesis by A3;
end;
