reserve E for non empty set;
reserve a for Element of E;
reserve A, B for Subset of E;
reserve Y for set;
reserve p for FinSequence;
reserve e, e1, e2 for Singleton of E;

theorem
  for E being finite non empty set, A,B,C being Event of E st 0 < prob(B
  /\ C) & 0 < prob(C) holds prob(A /\ B /\ C) = prob(A, B /\ C) * prob(B, C) *
  prob(C)
proof
  let E be finite non empty set, A,B,C be Event of E;
  assume that
A1: 0 < prob(B /\ C) and
A2: 0 < prob(C);
A3: prob(B /\ C) = prob(B, C) * prob(C) by A2,XCMPLX_1:87;
  prob(A /\ B /\ C) = prob(A /\ (B /\ C)) by XBOOLE_1:16;
  then prob(A /\ B /\ C) = prob(A, B /\ C) * prob(B /\ C) by A1,XCMPLX_1:87;
  hence thesis by A3;
end;
