reserve L for satisfying_DN_1 non empty ComplLLattStr;
reserve x, y, z for Element of L;

theorem Th58:
  for L being non empty ComplLLattStr, x, z being Element of L st L
  is join-commutative join-associative Huntington holds (z + x) *' (z + x`) = z
proof
  let L be non empty ComplLLattStr;
  let x, z be Element of L;
  assume L is join-commutative join-associative Huntington;
  then reconsider
  L9 = L as join-commutative join-associative Huntington non empty
  ComplLLattStr;
  reconsider z9 = z, x9 = x as Element of L9;
  (z9 + x9) *' (z9 + x9`) = z9 + (x9 *' x9`) by ROBBINS1:31
    .= z9 + Bot L9 by ROBBINS1:15
    .= z9 by ROBBINS1:13;
  hence thesis;
end;
