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,B3 being Event of E st 0 <
prob(B1) & 0 < prob(B2) & 0 < prob(B3) & B1 \/ B2 \/ B3 = E & B1 misses B2 & B1
  misses B3 & B2 misses B3 holds prob(B1, A) = ( prob(A, B1) * prob(B1) ) / ( (
  prob(A, B1) * prob(B1) + prob(A, B2) * prob(B2) ) + prob(A, B3) * prob(B3) )
proof
  let E be finite non empty set, A,B1,B2,B3 be Event of E;
  assume that
A1: 0 < prob(B1) and
A2: 0 < prob(B2) & 0 < prob(B3) & B1 \/ B2 \/ B3 = E & B1 misses B2 & B1
  misses B3 & B2 misses B3;
  prob(A) = ( prob(A, B1) * prob(B1) + prob(A, B2) * prob(B2) ) + prob(A,
  B3) * prob(B3) by A1,A2,Th51;
  hence thesis by A1,XCMPLX_1:87;
end;
