
theorem Th17:
  for L being join-commutative meet-commutative meet-absorbing
  join-absorbing join-idempotent distributive' non empty LattStr holds L is
  meet-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: ((x "/\" y) "/\" z) "\/" x = x by Th7;
A2: ((y "/\" z) "/\" x) "\/" y = (y "\/" (y "/\" z)) "/\" (y "\/" x) by Def5
    .= (y "\/" y) "/\" (y "\/" z) "/\" (y "\/" x) by Def5
    .= y "/\" (y "\/" z) "/\" (y "\/" x)
    .= y "/\" (y "\/" x) by LATTICES:def 9
    .= y by LATTICES:def 9;
  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 Def5
      .= ((x "/\" y) "\/" (x "/\" (y "/\" z))) "/\" z by Th7
      .= ((x "\/" (x "/\" (y "/\" z))) "/\" (y "\/" (x "/\" (y "/\" z))))
    "/\" z by Def5
      .= (x "/\" y) "/\" z by A2,LATTICES:def 8;
    A = (((x "/\" y) "/\" z) "\/" x) "/\" (((x "/\" y) "/\" z) "\/" (y
    "/\" z)) by Def5
      .= x "/\" ((((x "/\" y) "/\" z) "\/" y) "/\" (((x "/\" y) "/\" z) "\/"
    z)) by A1,Def5
      .= x "/\" (y "/\" (((x "/\" y) "/\" z) "\/" z)) by Th7
      .= x "/\" (y "/\" (((x "/\" y ) "\/" z) "/\" (z "\/" z))) by Def5
      .= x "/\" (y "/\" (((x "/\" y ) "\/" z) "/\" z))
      .= x "/\" (y "/\" z) by LATTICES:def 9;
    hence thesis by A3;
  end;
  hence thesis;
end;
