
theorem Th27: :: 4.31
  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`
  )
proof
  let L be join-commutative join-associative Huntington non empty
  ComplLLattStr, a, b, c be Element of L;
  set D = a *' b` *' c`, E = a` *' b *' c, F = a` *' b *' c`;
  set G = a` *' b` *' c, H = a` *' b` *' c`;
A1: a` = (a` *' b) + (a` *' b`) by Def6
    .= E + F + (a` *' b`) by Def6
    .= E + F + (G + H) by Def6;
A2: b` *' c` = (a *' (b` *' c`)) + (a` *' (b` *' c`)) by Th1
    .= D + (a` *' (b` *' c`)) by Th16
    .= D + H by Th16;
  (a *' (b + c))` = a` + (b + c)` by Th3
    .= a` + (b` *' c`)`` by Th17
    .= a` + (b` *' c`) by Th3;
  hence (a *' (b + c))` = E + F + (G + H) + H + D by A1,A2,LATTICES:def 5
    .= E + F + G + H + H + D by LATTICES:def 5
    .= E + F + G + (H + H) + D by LATTICES:def 5
    .= E + F + G + H + D by Def7
    .= D + (E + F + (G + H)) by LATTICES:def 5
    .= D + (E + F) + (G + H) by LATTICES:def 5
    .= D + (E + F) + G + H by LATTICES:def 5
    .= D + E + F + G + H by LATTICES:def 5;
end;
