
theorem Th26: :: 4.30
  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) + (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 A = a *' b *' c;
  a *' c = (a *' c *' b) + (a *' c *' b`) by Def6
    .= A + (a *' c *' b`) by Th16
    .= A + (a *' b` *' c) by Th16;
  hence (a *' b) + (a *' c) = A + (a *' b *' c`) + (A + (a *' b` *' c)) by Def6
    .= A + ((a *' b *' c`) + A) + (a *' b` *' c) by LATTICES:def 5
    .= A + A + (a *' b *' c`) + (a *' b` *' c) by LATTICES:def 5
    .= A + (a *' b *' c`) + (a *' b` *' c) by Def7;
end;
