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,B1,B2 being Event of E st 0 < prob
(B1) & 0 < prob(B2) & B1 \/ B2 = E & B1 misses B2 holds prob(B1, A) = ( prob(A,
  B1) * prob(B1) ) / ( prob(A, B1) * prob(B1) + prob(A, B2) * prob(B2) )
proof
  let E be finite non empty set, A,B1,B2 be Event of E;
  assume that
A1: 0 < prob(B1) and
A2: 0 < prob(B2) & B1 \/ B2 = E & B1 misses B2;
  prob(A) = prob(A, B1) * prob(B1) + prob(A, B2) * prob(B2) by A1,A2,Th50;
  hence thesis by A1,XCMPLX_1:87;
end;
