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;

theorem Th4:
  vol(M,FSets) is nonnegative
proof
  for n being Element of NAT holds 0 <= (vol(M,FSets)).n
  proof
    let n be Element of NAT;
    (vol(M,FSets)).n = M.(FSets.n) & {} in F by Def5,PROB_1:4;
    then M.{} <= (vol(M,FSets)).n by MEASURE1:8,XBOOLE_1:2;
    hence 0 <= (vol(M,FSets)).n by VALUED_0:def 19;
  end;
  hence thesis by SUPINF_2:39;
end;
