reserve
  A,B,X for set,
  S for SigmaField of X;
reserve C for C_Measure of X;

theorem Th15:
  for A being Subset of X holds C.A = 0. implies A in sigma_Field(C)
proof
  let A be Subset of X;
  assume
A1: C.A = 0.;
  now
    let W,Z be Subset of X;
    assume that
A2: W c= A and
    Z c= X \ A;
    Z c= W \/ Z by XBOOLE_1:7;
    then
A3: C.Z <= C.(W \/ Z) by Def1;
    C is nonnegative by Def1;
    then 0.<= C.W by MEASURE1:def 2;
    then C.W = 0. by A1,A2,Def1;
    hence C.W + C.Z <= C.(W \/ Z) by A3,XXREAL_3:4;
  end;
  hence thesis by Def2;
end;
