
theorem Th5:
  for X being set, a being Element of BoolePoset X holds 'not' a = X \ a
proof
  let X be set, a be Element of BoolePoset X;
A1: the carrier of BoolePoset X = bool X by Th2;
  reconsider b = X\a as Element of BoolePoset X by Th2;
A2: a misses b by XBOOLE_1:79;
A3: a"/\"b = a /\ b by YELLOW_1:17
    .= {} by A2
    .= Bottom BoolePoset X by YELLOW_1:18;
  a"\/"b = a \/ b by YELLOW_1:17
    .= X by A1,XBOOLE_1:45
    .= Top BoolePoset X by YELLOW_1:19;
  then b is_a_complement_of a by A3,WAYBEL_1:def 23;
  hence thesis by YELLOW_5:11;
end;
