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

theorem Th5:
  for A being Subset of X holds (A in sigma_Field(C) iff for W,Z
  being Subset of X holds (W c= A & Z c= X \ A implies C.W + C.Z = C.(W \/ Z)))
proof
  let A be Subset of X;
  hereby
    assume
A1: A in sigma_Field(C);
    let W,Z be Subset of X;
    assume that
A2: W c= A and
A3: Z c= X \ A;
A4: C.(W \/ Z) <= C.W + C.Z by Th4;
    C.W + C.Z <= C.(W \/ Z) by A1,A2,A3,Def2;
    hence C.W + C.Z = C.(W \/ Z) by A4,XXREAL_0:1;
  end;
  assume for W,Z being Subset of X holds (W c= A & Z c= X \ A implies C.W + C
  .Z = C.(W \/ Z));
  then for W,Z being Subset of X holds (W c= A & Z c= X \ A implies C.W + C.Z
  <= C.(W \/ Z));
  hence thesis by Def2;
end;
