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 Th29:
  for P,B,A st 0 < P.B holds P.(A /\ B) = P.|.B.A * P.B
proof
  let P,B,A;
  assume
A1: 0 < P.B;
  then P.|.B.A * P.B = (P.(A /\ B)/P.B) * P.B by Def6
    .= P.(A /\ B) * (P.B)" * P.B by XCMPLX_0:def 9
    .= P.(A /\ B) * ((P.B)" * P.B)
    .= P.(A /\ B) * 1 by A1,XCMPLX_0:def 7
    .= P.(A /\ B);
  hence thesis;
end;
