theorem
  P.(A \/ B \/ C) = P.A + P.B + P.C - (P.(A /\ B) + P.(B /\ C) + P.(A /\
  C)) + P.(A /\ B /\ C)
proof
A1: P.((A \/ B) /\ C) = P.((A /\ C) \/ (B /\ C)) by XBOOLE_1:23
    .= P.(A /\ C) + P.(B /\ C) - P.((A /\ C) /\ (B /\ C)) by PROB_1:38
    .= P.(A /\ C) + P.(B /\ C) - P.(A /\ ((B /\ C) /\ C)) by XBOOLE_1:16
    .= P.(A /\ C) + P.(B /\ C) - P.(A /\ (B /\ (C /\ C))) by XBOOLE_1:16
    .= P.(B /\ C) + P.(A /\ C) - P.(A /\ B /\ C) by XBOOLE_1:16;
  P.(A \/ B \/ C) = P.(A \/ B) + P.C - P.((A \/ B) /\ C) by PROB_1:38
    .= P.A + P.B - P.(A /\ B) + P.C - P.((A \/ B) /\ C) by PROB_1:38
    .= P.A + P.B + P.C - (P.(A /\ B) + P.((A \/ B) /\ C));
  hence thesis by A1;
end;
