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 Th31:
  for P,A,B,C st C = B` & 0 < P.B & 0 < P.C holds P.A = P.|.B.A *
  P.B + P.|.C.A * P.C
proof
  let P,A,B,C;
  assume that
A1: C = B` and
A2: 0 < P.B and
A3: 0 < P.C;
  P.A = P.(A /\ B) + P.(A /\ C) by A1,Th14
    .= P.|.B.A * P.B + P.(A /\ C) by A2,Th29
    .= P.|.B.A * P.B + P.|.C.A * P.C by A3,Th29;
  hence thesis;
end;
