reserve X for set,
  F for Field_Subset of X,
  M for Measure of F,
  A,B for Subset of X,
  Sets for SetSequence of X,
  seq,seq1,seq2 for ExtREAL_sequence,
  n,k for Nat;
reserve FSets for Set_Sequence of F,
  CA for Covering of A,F;
reserve Cvr for Covering of Sets,F;
reserve C for C_Measure of X;

theorem Th19:
  (for B being Subset of X holds C.(B /\ A) + C.(B /\ (X \ A)) <=
  C.B) implies A in sigma_Field C
proof
  assume
A1: for B being Subset of X holds C.(B /\ A) + C.(B /\ (X \ A)) <= C.B;
  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 W,Z be Subset of X;
    assume that
A2: W c= A and
A3: Z c= X \ A;
    set Y = W \/ Z;
A4: Z misses A by A3,XBOOLE_1:106;
A5: Y /\ (X \ A) = (Y /\ X) \ A by XBOOLE_1:49
      .= (X /\ W \/ X /\ Z) \ A by XBOOLE_1:23
      .= (W \/ X /\ Z) \ A by XBOOLE_1:28
      .= (W \/ Z) \ A by XBOOLE_1:28
      .= (W \ A) \/ (Z \ A) by XBOOLE_1:42
      .= {} \/ (Z \ A) by A2,XBOOLE_1:37
      .= {} \/ Z by A4,XBOOLE_1:83;
    Y /\ A = A /\ W \/ A /\ Z by XBOOLE_1:23
      .= W \/ A /\ Z by A2,XBOOLE_1:28
      .= W \/ {} by A4,XBOOLE_0:def 7;
    hence thesis by A1,A5;
  end;
  hence A in sigma_Field C by MEASURE4:def 2;
end;
