
theorem
for X be set, S be with_empty_element semi-diff-closed cap-closed
      Subset-Family of X,
    P be pre-Measure of S
 ex M be Function of Ring_generated_by S,ExtREAL st
  M.{} = 0 &
  for K be disjoint_valued FinSequence of S
    st Union K in Ring_generated_by S holds M.(Union K) = Sum(P*K)
proof
   let X be set, S be with_empty_element semi-diff-closed cap-closed
       Subset-Family of X,
       P be pre-Measure of S;
   consider M be nonnegative additive zeroed Function
                of (Ring_generated_by S),ExtREAL such that
A1: for A be set st A in Ring_generated_by S holds
     for F be disjoint_valued FinSequence of S st
      A = Union F holds M.A = Sum(P*F) by Th46;
   take M;
   thus M.{} = 0 by VALUED_0:def 19;
   thus for K be disjoint_valued FinSequence of S
     st Union K in Ring_generated_by S holds M.(Union K) = Sum(P*K) by A1;
end;
