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 Th14:
  C = B` implies P.A = P.(A /\ B) + P.(A /\ C)
proof
  assume
A1: C = B`;
  then B misses C by SUBSET_1:24;
  then A /\ C misses B by XBOOLE_1:74;
  then
A2: A /\ B misses A /\ C by XBOOLE_1:74;
  P.A = P.(A /\ [#]Omega) by XBOOLE_1:28
    .= P.(A /\ (B \/ C)) by A1,SUBSET_1:10
    .= P.(A /\ B \/ A /\ C) by XBOOLE_1:23
    .= P.(A /\ B) + P.(A /\ C) by A2,PROB_1:def 8;
  hence thesis;
end;
