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 Th50:
  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(A) = 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) and
A3: B1 \/ B2 = E and
A4: B1 misses B2;
A5: B2 \ B1 = E \ B1 by A3,XBOOLE_1:40;
  then 0 < prob((B1)`) by A2,A4,XBOOLE_1:83;
  then 0 < 1 - prob(B1) by Th22;
  then
A6: 1 - ( 1 - prob(B1) ) < 1 by XREAL_1:44;
  B2 = B1` by A4,A5,XBOOLE_1:83;
  hence thesis by A1,A6,Th49;
end;
