reserve X for set;

theorem
  for S being Field_Subset of X, M being Measure of S, A,B being
measure_zero of M holds A \/ B is measure_zero of M & A /\ B is measure_zero of
  M & A \ B is measure_zero of M
proof
  let S be Field_Subset of X, M be Measure of S, A,B be measure_zero of M;
A1: 0.<= M.(A /\ B) by Def2;
A2: 0.<= M.(A \ B) by Def2;
A3: M.A = 0. by Def4;
  then M.(A \ B) <= 0.by Th8,XBOOLE_1:36;
  then
A4: M.(A \ B) = 0.by A2;
  M.B = 0. by Def4;
  then M.(A \/ B) <= 0.+ 0.by A3,Th10;
  then
A5: M.(A \/ B) <= 0.by XXREAL_3:4;
  0.<= M.(A \/ B) by Def2;
  then
A6: M.(A \/ B) = 0.by A5;
  M.(A /\ B) <= 0.by A3,Th8,XBOOLE_1:17;
  then M.(A /\ B) = 0.by A1;
  hence thesis by A6,A4,Def4;
end;
