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

theorem Th4:
  for W,Z being Subset of X holds C.(W \/ Z) <= C.W + C.Z
proof
  let W,Z be Subset of X;
  reconsider P = {} as Subset of X by XBOOLE_1:2;
  consider F being sequence of bool X such that
A1: rng F = {W,Z,P} and
A2: F.0 = W and
A3: F.1 = Z and
A4: for n being Element of NAT st 1 < n holds F.n = P by MEASURE1:17;
A5: (C*F).1 = C.Z by A3,FUNCT_2:15;
  set G = C*F;
A6: union {W,Z,P} = W \/ Z
  proof
    thus union {W,Z,P} c= W \/ Z
    proof
      let x be object;
      assume x in union {W,Z,P};
      then ex Y being set st x in Y & Y in {W,Z,P} by TARSKI:def 4;
      then x in W or x in Z or x in P by ENUMSET1:def 1;
      hence thesis by XBOOLE_0:def 3;
    end;
    let x be object;
    assume
A7: x in W \/ Z;
    now
      per cases by A7,XBOOLE_0:def 3;
      suppose
A8:     x in W;
        take Y = W;
        thus x in Y & Y in {W,Z,P} by A8,ENUMSET1:def 1;
      end;
      suppose
A9:     x in Z;
        take Y = Z;
        thus x in Y & Y in {W,Z,P} by A9,ENUMSET1:def 1;
      end;
    end;
    hence thesis by TARSKI:def 4;
  end;
A10: C is zeroed by Def1;
A11: for r being Element of NAT st 2 <= r holds (C*F).r = 0.
  proof
    let r be Element of NAT;
    assume 2 <= r;
    then 1 + 1 <= r;
    then 1 < r by NAT_1:13;
    then
A12: F.r = {} by A4;
    C is zeroed by Def1;
    then C.(F.r) = 0. by A12,VALUED_0:def 19;
    hence thesis by FUNCT_2:15;
  end;
  C is nonnegative by Def1;
  then
A13: C*F is nonnegative by MEASURE1:25;
  F.2 = P by A4;
  then
A14: (C*F).2 = C.P by FUNCT_2:15;
A15: (C*F).0 = C.W by A2,FUNCT_2:15;
  consider y1,y2 being R_eal such that
A16: y1 = Ser(G).1 and
A17: y2 = Ser(G).0;
  Ser(G).2 = y1 + G.(1 + 1) by A16,SUPINF_2:def 11
    .= Ser(G).1 + 0. by A14,A16,A10,VALUED_0:def 19
    .= Ser(G).1 by XXREAL_3:4
    .= y2 + G.(0 + 1) by A17,SUPINF_2:def 11
    .= C.W + C.Z by A5,A15,A17,SUPINF_2:def 11;
  then SUM(C*F) = C.W + C.Z by A13,A11,SUPINF_2:48;
  hence thesis by A1,A6,Def1;
end;
