
theorem Th11: :: 4.15 proof without join-idempotency, no top at all
  for L being join-commutative join-associative Huntington non
  empty ComplLLattStr, a, b being Element of L holds a + (b + b`)` = a
proof
  let L be join-commutative join-associative Huntington non empty
  ComplLLattStr, a, b be Element of L;
  set O = b + b`;
A1: O`` = O by Th3;
A2: O` = (O`` + O``)` + (O`` + O`)` by Def6
    .= (O + O)` + O` by A1,Th4;
A3: O = a` + a by Th4;
  O = O + O` by Th4
    .= O + O` + (O + O)` by A2,LATTICES:def 5
    .= O + (O + O)` by Th4;
  then
A4: O + O = O + O + (O + O)` by LATTICES:def 5
    .= O by Th4;
  hence a + O` = ((a` + a`)` + (a` + a)`) + ((a` + a)` + (a` + a)`) by A2,A3
,Def6
    .= (a` + a`)` + ((a` + a)` + (a` + a)`) by A2,A4,A3,LATTICES:def 5
    .= a by A2,A4,A3,Def6;
end;
