reserve Omega for set;
reserve m,n,k for Nat;
reserve x,y for object;
reserve r,r1,r2,r3 for Real;
reserve seq,seq1 for Real_Sequence;
reserve Sigma for SigmaField of Omega;
reserve ASeq,BSeq for SetSequence of Sigma;
reserve A, B, C, A1, A2, A3 for Event of Sigma;
reserve Omega for non empty set;
reserve Sigma for SigmaField of Omega;
reserve A, B, C, A1, A2, A3 for Event of Sigma;
reserve ASeq,BSeq for SetSequence of Sigma;
reserve P,P1,P2 for Probability of Sigma;

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;
