
theorem Th5:
  for L being non empty LattStr holds L is Lattice-like iff L is
  meet-Associative join-Associative meet-Absorbing join-absorbing
proof
  let L be non empty LattStr;
A1: L is Lattice-like implies L is meet-Associative join-Associative
  meet-Absorbing join-absorbing
  proof
    assume
A2: L is Lattice-like;
A3: for x,y,z being Element of L holds x "/\" (y "/\" z) = y "/\" (x "/\" z)
    proof
      let a,b,c be Element of L;
      a "/\" (b "/\" c) = (a "/\" b) "/\" c by A2,LATTICES:def 7
        .= (b "/\" a) "/\" c by A2,LATTICES:def 6
        .= b "/\" (a "/\" c) by A2,LATTICES:def 7;
      hence thesis;
    end;
A4: for x,y being Element of L holds x "\/" (x "/\" y) = x
    proof
      let a,b be Element of L;
      a = (b "/\" a) "\/" a by A2,LATTICES:def 8
        .= (a "/\" b) "\/" a by A2,LATTICES:def 6
        .= a "\/" (a "/\" b) by A2,LATTICES:def 4;
      hence thesis;
    end;
    for x,y,z being Element of L holds x "\/" (y "\/" z) = y "\/" (x "\/" z)
    proof
      let a,b,c be Element of L;
      a "\/" (b "\/" c) = (a "\/" b) "\/" c by A2,LATTICES:def 5
        .= (b "\/" a) "\/" c by A2,LATTICES:def 4
        .= b "\/" (a "\/" c) by A2,LATTICES:def 5;
      hence thesis;
    end;
    hence thesis by A2,A3,A4;
  end;
  L is meet-Associative join-Associative meet-Absorbing join-absorbing
  implies L is Lattice-like
  by Th3,Th2,Th4;
  hence thesis by A1;
end;
