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 Th5:
  0 <= (Volume(M,Cvr)).n
proof
  for k being Element of NAT holds 0 <= (vol(M,Cvr.n)).k
  proof
    let k be Element of NAT;
    0 <= M.((Cvr.n).k) by SUPINF_2:51;
    hence thesis by Def5;
  end;
  then
A1: vol(M,Cvr.n) is nonnegative by SUPINF_2:39;
  (Volume(M,Cvr)).n = SUM(vol(M,Cvr.n)) by Def6;
  hence thesis by A1,MEASURE6:2;
end;
