
theorem Th25: :: 4.29
  for L being join-commutative join-associative Huntington non
  empty ComplLLattStr, a, b, c being Element of L
   holds (a *' c) *' (b *' c`) =
Bot L & (a *' b *' c) *' (a` *' b *' c) = Bot L & (a *' b` *' c) *' (a` *' b *'
  c) = Bot L & (a *' b *' c) *' (a` *' b` *' c) = Bot L & (a *' b *' 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;
A1: for a, b, c being Element of L holds (a *' c) *' (b *' c`) = Bot L
  proof
    let a, b, c be Element of L;
    thus (a *' c) *' (b *' c`) = (a *' c) *' c` *' b by Th16
      .= a *' (c *' c`) *' b by Th16
      .= a *' Bot L *' b by Th15
      .= Bot L *' b by Def9
      .= Bot L by Def9;
  end;
  hence (a *' c) *' (b *' c`) = Bot L;
  thus a *' b *' c *' (a` *' b *' c) = a *' (b *' c) *' (a` *' b *' c) by Th16
    .= b *' c *' a *' (a` *' b) *' c by Th16
    .= b *' c *' a *' a` *' b *' c by Th16
    .= b *' c *' (a *' a`) *' b *' c by Th16
    .= b *' c *' (a *' a`) *' (b *' c) by Th16
    .= b *' c *' Bot L *' (b *' c) by Th15
    .= Bot L *' (b *' c) by Def9
    .= Bot L by Def9;
  thus (a *' b` *' c) *' (a` *' b *' c) = a *' (b` *' c) *' (a` *' b *' c) by
Th16
    .= (b` *' c) *' a *' (a` *' (b *' c)) by Th16
    .= (b` *' c) *' a *' a` *' (b *' c) by Th16
    .= (b` *' c) *' (a *' a`) *' (b *' c) by Th16
    .= (b` *' c) *' Bot L *' (b *' c) by Th15
    .= Bot L *' (b *' c) by Def9
    .= Bot L by Def9;
  thus (a *' b *' c) *' (a` *' b` *' c) = (a *' (b *' c)) *' (a` *' b` *' c)
  by Th16
    .= (a *' (b *' c)) *' (a` *' (b` *' c)) by Th16
    .= Bot L by A1;
  thus (a *' b *' c`) *' (a` *' b` *' c`) = (a *' (b *' c`)) *' (a` *' b` *' c
  `) by Th16
    .= (a *' (b *' c`)) *' (a` *' (b` *' c`)) by Th16
    .= Bot L by A1;
end;
