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 Th15:
  P.A + P.B - 1 <= P.(A /\ B)
proof
  P.A + P.B - P.(A /\ B) = P.(A \/ B) by PROB_1:38;
  then P.A + P.B - P.(A /\ B) <= 1 by PROB_1:35;
  then P.A + P.B <= P.(A /\ B) + 1 by XREAL_1:20;
  hence thesis by XREAL_1:20;
end;
