reserve X for set;

theorem Th9:
  for S being Field_Subset of X, M being Measure of S,
      A,B being Element of S st A c= B & M.A < +infty holds
    M.(B \ A) = M.B - M.A
proof
  let S be Field_Subset of X, M be Measure of S, A,B be Element of S;
  set C = B \ A;
  assume that
A1: A c= B and
A2: M.A < +infty;
A3: 0.<= M.A by Def2;
  A misses C by XBOOLE_1:79;
  then M.(A \/ C) = M.A + M.C by Def3;
  then M.B = M.A + M.C by A1,XBOOLE_1:45;
  hence thesis by A2,A3,XXREAL_3:28;
end;
