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
  for A,B,C,P st A,B are_independent_respect_to P & A,C
  are_independent_respect_to P & B misses C holds A,B \/ C
  are_independent_respect_to P
proof
  let A,B,C,P;
  assume that
A1: A,B are_independent_respect_to P and
A2: A,C are_independent_respect_to P and
A3: B misses C;
A4: (A /\ B) misses (A /\ C) by A3,XBOOLE_1:76;
  P.(A /\ (B \/ C)) = P.((A /\ B) \/ (A /\ C)) by XBOOLE_1:23
    .= P.(A /\ B) + P.(A /\ C) by A4,PROB_1:def 8
    .= P.A * P.B + P.(A /\ C) by A1
    .= P.A * P.B + P.A * P.C by A2
    .= P.A * (P.B + P.C)
    .= P.A * P.(B \/ C) by A3,PROB_1:def 8;
  hence thesis;
end;
