
theorem Th24: :: 4.28
  for L being join-commutative join-associative Huntington non
  empty ComplLLattStr, a, b, c being Element of L
   holds (a *' b *' c) + (a *' b
*' c`) + (a *' b` *' c) + (a *' b` *' c`) + (a` *' b *' c) + (a` *' b *' c`) +
  (a` *' b` *' 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`;
  A + B + C + D + E + F + G + H = (a *' b) + C + D + E + F + G + H by Def6
    .= (a *' b) + (C + D) + E + F + G + H by LATTICES:def 5
    .= (a *' b) + (a *' b`) + E + F + G + H by Def6
    .= (a *' b) + (a *' b`) + (E + F) + G + H by LATTICES:def 5
    .= (a *' b) + (a *' b`) + (a` *' b) + G + H by Def6
    .= (a *' b) + (a *' b`) + (a` *' b) + (G + H) by LATTICES:def 5
    .= (a *' b) + (a *' b`) + (a` *' b) + (a` *' b`) by Def6
    .= a + (a` *' b) + (a` *' b`) by Def6
    .= a + ((a` *' b) + (a` *' b`)) by LATTICES:def 5
    .= a + a` by Def6;
  hence thesis by Def8;
end;
