reserve X for set;

theorem Th8:
  for S being Field_Subset of X, M being Measure of S, A,B being
  Element of S st A c= B holds M.A <= M.B
proof
  let S be Field_Subset of X, M be Measure of S, A,B be Element of S;
  reconsider C = B \ A as Element of S;
A1: 0.<= M.C by Def2;
  A misses C by XBOOLE_1:79;
  then
A2: M.(A \/ C) = M.A + M.C by Def3;
  assume A c= B;
  then M.B = M.A + M.C by A2,XBOOLE_1:45;
  hence thesis by A1,XXREAL_3:39;
end;
