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 Th26:
  for E being finite non empty set, A,B being Event of E holds
  prob(A) = prob(A /\ B) + prob(A /\ B`)
proof
  let E be finite non empty set, A,B be Event of E;
  A = A /\ ( A \/ [#] E ) by XBOOLE_1:21;
  then A = A /\ [#] E by SUBSET_1:11;
  then
A1: A = A /\ ( B \/ B`) by SUBSET_1:10;
  prob((A /\ B) \/ (A /\ B`)) = prob(A /\ B) + prob(A /\ B`) by Th13,Th21;
  hence thesis by A1,XBOOLE_1:23;
end;
