
theorem Th5:
  for L being complemented' join-commutative meet-commutative
  join-idempotent distributive upper-bounded' lower-bounded' distributive' non
  empty LattStr for x being Element of L holds x "/\" Bot' L = Bot' L
proof
  let L be complemented' join-commutative meet-commutative join-idempotent
  distributive upper-bounded' lower-bounded' distributive' non empty LattStr;
  let x be Element of L;
  x "/\" Bot' L = (x "/\" Bot' L) "\/" Bot' L by Def4
    .= (x "/\" Bot' L) "\/" (x "/\" x`# ) by Th3
    .= x "/\" (Bot' L "\/" x`# ) by LATTICES:def 11
    .= x "/\" x`# by Def4
    .= Bot' L by Th3;
  hence thesis;
end;
