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 P,A,B,C st 0 < P.(A /\ B) holds P.(A /\ B /\ C) = P.A * P.|.A.B *
  P.|.(A /\ B).C
proof
  let P,A,B,C;
  assume
A1: 0 < P.(A /\ B);
  then
A2: 0 < P.A by PROB_1:34,XBOOLE_1:17;
  P.(A /\ B /\ C) = P.(B /\ A) * P.|.(A /\ B).C by A1,Th29
    .= P.A * P.|.A.B * P.|.(A /\ B).C by A2,Th29;
  hence thesis;
end;
