
theorem Th28: :: 4.32
  for L being join-commutative join-associative Huntington non
empty ComplLLattStr, a, b, c being Element of L
 holds ((a *' b) + (a *' c)) + (a *' (b + c))` = Top L
proof
  let L be join-commutative join-associative Huntington non empty
  ComplLLattStr, a, b, c be Element of L;
  set A = a *' b *' c, B = a *' b *' c`, C = a *' b` *' c;
  set D = a *' b` *' c`, E = a` *' b *' c, F = a` *' b *' c`;
  set G = a` *' b` *' c, H = a` *' b` *' c`;
  set ABC = A + B + C, GH = G + H;
  (a *' (b + c))` = D + E + F + G + H & (a *' b) + (a *' c) = ABC by Th26,Th27;
  then (a *' b) + (a *' c) + (a *' (b + c))` = ABC + (D + E + F + GH) by
LATTICES:def 5
    .= ABC + (D + E + (F + GH)) by LATTICES:def 5
    .= ABC + (D + E) + (F + GH) by LATTICES:def 5
    .= ABC + D + E + (F + GH) by LATTICES:def 5
    .= ABC + D + (E + (F + GH)) by LATTICES:def 5
    .= ABC + D + (E + (F + G + H)) by LATTICES:def 5
    .= ABC + D + E + (F + G + H) by LATTICES:def 5
    .= ABC + D + E + (F + GH) by LATTICES:def 5
    .= ABC + D + E + F + GH by LATTICES:def 5
    .= ABC + D + E + F + G + H by LATTICES:def 5
    .= Top L by Th24;
  hence thesis;
end;
