reserve X for set;

theorem
  for S being SigmaField of X, M being sigma_Measure of S, A being
  Element of S, B being measure_zero of M holds M.(A \/ B) = M.A & M.(A /\ B) =
  0.& M.(A \ B) = M.A
proof
  let S be SigmaField of X, M be sigma_Measure of S, A be Element of S, B be
  measure_zero of M;
A1: M.A = M.((A /\ B) \/ (A \ B)) by XBOOLE_1:51;
A2: M.B = 0.by Def7;
  then
A3: M.(A /\ B) <= 0.by Th8,XBOOLE_1:17;
A4: 0.<= M.(A /\ B) by Def2;
  then M.(A /\ B) = 0.by A3;
  then M.A <= 0.+ M.(A \ B) by A1,Th10;
  then
A5: M.A <= M.(A \ B) by XXREAL_3:4;
  M.(A \/ B) <= M.A + 0.by A2,Th10;
  then
A6: M.(A \/ B) <= M.A by XXREAL_3:4;
  M.A <= M.(A \/ B) & M.(A \ B) <= M.A by Th8,XBOOLE_1:7,36;
  hence thesis by A4,A6,A3,A5,XXREAL_0:1;
end;
