reserve X for set;

theorem
  for S being SigmaField of X, M being Measure of S st (for F being
  Sep_Sequence of S holds M.(union rng F) <= SUM(M*F)) holds M is sigma_Measure
  of S
proof
  let S be SigmaField of X, M be Measure of S;
  assume
A1: for F being Sep_Sequence of S holds M.(union rng F) <= SUM(M*F);
  for F being Sep_Sequence of S holds SUM(M*F) = M.(union rng F)
  proof
    let F be Sep_Sequence of S;
    M.(union rng F) <= SUM(M*F) & SUM(M*F) <= M.(union rng F) by A1,Th13;
    hence thesis by XXREAL_0:1;
  end;
  hence thesis by MEASURE1:def 6;
end;
