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;
