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,B being Event of E st A, B
  are_independent holds A`, B are_independent
proof
  let E be finite non empty set, A,B be Event of E;
  prob(A` /\ B) = prob(B \ A) by SUBSET_1:13;
  then
A1: prob(A` /\ B) = prob(B) - prob(A /\ B) by Th23;
  assume A, B are_independent;
  then prob(A` /\ B) = 1 * prob(B) - prob(A) * prob(B) by A1;
  then prob(A` /\ B) = ( 1 - prob(A) ) * prob(B);
  then prob(A` /\ B) = prob(A`) * prob(B) by Th22;
  hence thesis;
end;
