
theorem Th29: :: 4.33
  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))` = Bot 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 DEFG = D + E + F + G;
  (A *' D) + (A *' E) = Bot L + (A *' E) by Th25
    .= Bot L + Bot L by Th25
    .= Bot L by Def7;
  then Top L = Bot L + (A *' (D + E))` by Th28
    .= (A *' (D + E))` by Th13;
  then Bot L = (A *' (D + E))`` by Th9
    .= A *' (D + E) by Th3;
  then (A *' (D + E)) + (A *' F) = Bot L + Bot L by Th25
    .= Bot L by Def7;
  then Top L = Bot L + (A *' (D + E + F))` by Th28
    .= (A *' (D + E + F))` by Th13;
  then
A1: Bot L = (A *' (D + E + F))`` by Th9
    .= A *' (D + E + F) by Th3;
  A *' G = Bot L by Th25;
  then (A *' (D + E + F)) + (A *' G) = Bot L by A1,Def7;
  then Top L = Bot L + (A *' DEFG)` by Th28
    .= (A *' DEFG)` by Th13;
  then Bot L = (A *' DEFG)`` by Th9
    .= A *' DEFG by Th3;
  then (A *' DEFG) + (A *' H) = Bot L + Bot L by Th25
    .= Bot L by Def7;
  then
A2: Top L = Bot L + (A *' (DEFG + H))` by Th28
    .= (A *' (DEFG + H))` by Th13;
  (B *' D) + (B *' E) = Bot L + (B *' E) by Th25
    .= Bot L + Bot L by Th25
    .= Bot L by Def7;
  then Top L = Bot L + (B *' (D + E))` by Th28
    .= (B *' (D + E))` by Th13;
  then Bot L = (B *' (D + E))`` by Th9
    .= B *' (D + E) by Th3;
  then (B *' (D + E)) + (B *' F) = Bot L + Bot L by Th25
    .= Bot L by Def7;
  then Top L = Bot L + (B *' (D + E + F))` by Th28
    .= (B *' (D + E + F))` by Th13;
  then Bot L = (B *' (D + E + F))`` by Th9
    .= B *' (D + E + F) by Th3;
  then (B *' (D + E + F)) + (B *' G) = Bot L + Bot L by Th25
    .= Bot L by Def7;
  then Top L = Bot L + (B *' DEFG)` by Th28
    .= (B *' DEFG)` by Th13;
  then
A3: Bot L = (B *' DEFG)`` by Th9
    .= B *' DEFG by Th3;
  C *' D = Bot L by Th25;
  then (C *' D) + (C *' E) = Bot L + Bot L by Th25
    .= Bot L by Def7;
  then Top L = Bot L + (C *' (D + E))` by Th28
    .= (C *' (D + E))` by Th13;
  then Bot L = (C *' (D + E))`` by Th9
    .= C *' (D + E) by Th3;
  then (C *' (D + E)) + (C *' F) = Bot L + Bot L by Th25
    .= Bot L by Def7;
  then Top L = Bot L + (C *' (D + E + F))` by Th28
    .= (C *' (D + E + F))` by Th13;
  then Bot L = (C *' (D + E + F))`` by Th9
    .= C *' (D + E + F) by Th3;
  then (C *' (D + E + F)) + (C *' G) = Bot L + Bot L by Th25
    .= Bot L by Def7;
  then Top L = Bot L + (C *' DEFG)` by Th28
    .= (C *' DEFG)` by Th13;
  then Bot L = (C *' DEFG)`` by Th9
    .= C *' DEFG by Th3;
  then (C *' DEFG) + (C *' H) = Bot L + Bot L by Th25
    .= Bot L by Def7;
  then
A4: Top L = Bot L + (C *' (DEFG + H))` by Th28
    .= (C *' (DEFG + H))` by Th13;
  B *' H = Bot L by Th25;
  then (B *' DEFG) + (B *' H) = Bot L by A3,Def7;
  then
A5: Top L = Bot L + (B *' (DEFG + H))` by Th28
    .= (B *' (DEFG + H))` by Th13;
A6: B *' (DEFG + H) = (B *' (DEFG + H))`` by Th3
    .= Bot L by A5,Th9;
  A *' (DEFG + H) = (A *' (DEFG + H))`` by Th3
    .= Bot L by A2,Th9;
  then (A *' (DEFG + H)) + (B *' (DEFG + H)) = Bot L by A6,Def7;
  then Top L = Bot L + ((A + B) *' (DEFG + H))` by Th28
    .= ((A + B) *' (DEFG + H))` by Th13;
  then
A7: Bot L = ((A + B) *' (DEFG + H))`` by Th9
    .= (A + B) *' (DEFG + H) by Th3;
  C *' (DEFG + H) = (C *' (DEFG + H))`` by Th3
    .= Bot L by A4,Th9;
  then ((A + B) *' (DEFG + H)) + (C *' (DEFG + H)) = Bot L by A7,Def7;
  then Top L = Bot L + ((A + B + C) *' (DEFG + H))` by Th28
    .= ((A + B + C) *' (DEFG + H))` by Th13;
  then
A8: Bot L = ((A + B + C) *' (DEFG + H))`` by Th9
    .= (A + B + C) *' (DEFG + H) by Th3;
  (a *' (b + c))` = DEFG + H by Th27;
  hence thesis by A8,Th26;
end;
