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 Th20:
  for E being finite non empty set, A,B being Event of E holds
  prob(A \/ B) = prob(A) + prob(B) - prob(A /\ B)
proof
  let E be finite non empty set, A,B be Event of E;
  set q = ( card E )";
  set p = card E;
  card (( A \/ B ) qua Event of E) = card A + card B - card ( A /\ B ) by
CARD_2:45;
  then
  card ( A \/ B ) * q = card A * q + ( card B * q - card ( A /\ B ) * q );
  then card ( A \/ B ) / p = card A * q + card B * q - card ( A /\ B ) * q by
XCMPLX_0:def 9;
  then card ( A \/ B ) / p = card A / p + card B * q - card ( A /\ B ) * q by
XCMPLX_0:def 9;
  then card ( A \/ B ) / p = card A / p + card B / p - card ( A /\ B ) * q by
XCMPLX_0:def 9;
  hence thesis by XCMPLX_0:def 9;
end;
