
theorem Th16:
  for L being join-commutative meet-commutative meet-absorbing
  join-absorbing join-idempotent distributive non empty LattStr holds L is
  join-associative
proof
  let L be join-commutative meet-commutative meet-absorbing join-absorbing
  join-idempotent distributive non empty LattStr;
  let x, y, z be Element of L;
A1: ((y "\/" z) "\/" x) "/\" y = (y "/\" (y "\/" z)) "\/" (y "/\" x) by
LATTICES:def 11
    .= (y "/\" y) "\/" (y "/\" z) "\/" (y "/\" x) by LATTICES:def 11
    .= y "\/" (y "/\" z) "\/" (y "/\" x)
    .= y "\/" (y "/\" x) by LATTICES:def 8
    .= y by LATTICES:def 8;
A2: ((x "\/" y) "\/" z) "/\" x = x by Th6;
  x "\/" (y "\/" z) = (x "\/" y) "\/" z
  proof
    set A = ((x "\/" y) "\/" z) "/\" (x "\/" (y "\/" z));
A3: A = ((x "\/" y) "/\" (x "\/" (y "\/" z))) "\/" (z "/\" (x "\/" (y "\/"
    z))) by LATTICES:def 11
      .= ((x "\/" y) "/\" (x "\/" (y "\/" z))) "\/" z by Th6
      .= ((x "/\" (x "\/" (y "\/" z))) "\/" (y "/\" (x "\/" (y "\/" z))))
    "\/" z by LATTICES:def 11
      .= (x "\/" y) "\/" z by A1,LATTICES:def 9;
    A = (((x "\/" y) "\/" z) "/\" x) "\/" (((x "\/" y) "\/" z) "/\" (y
    "\/" z)) by LATTICES:def 11
      .= x "\/" ((((x "\/" y) "\/" z) "/\" y) "\/" (((x "\/" y) "\/" z) "/\"
    z)) by A2,LATTICES:def 11
      .= x "\/" (y "\/" (((x "\/" y) "\/" z) "/\" z)) by Th6
      .= x "\/" (y "\/" (((x "\/" y ) "/\" z) "\/" (z "/\" z))) by
LATTICES:def 11
      .= x "\/" (y "\/" (((x "\/" y ) "/\" z) "\/" z))
      .= x "\/" (y "\/" z) by LATTICES:def 8;
    hence thesis by A3;
  end;
  hence thesis;
end;
