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 Th49:
  for E being finite non empty set, A,B being Event of E st 0 <
prob(B) & prob(B) < 1 holds prob(A) = prob(A, B) * prob(B) + prob(A, B`) * prob
  (B`)
proof
  let E be finite non empty set, A,B be Event of E;
  assume that
A1: 0 < prob(B) and
A2: prob(B) < 1;
  prob(B) -1 < 1 - 1 by A2,XREAL_1:9;
  then 0 < - ( - ( 1 - prob(B) ) );
  then
A3: 0 < prob(B`) by Th22;
  prob(A) = prob(A /\ B) + prob(A /\ B`) by Th26;
  then prob(A) = prob(A, B) * prob(B) + prob(A /\ B`) by A1,XCMPLX_1:87;
  hence thesis by A3,XCMPLX_1:87;
end;
