
theorem Th12:
  for A being Subset of REAL holds for F being Interval_Covering
  of A holds (F vol) is nonnegative
proof
  let A be Subset of REAL;
  let F be Interval_Covering of A;
  for n being Element of NAT holds 0. <= (F vol).n
  proof
    let n be Element of NAT;
    (F vol).n = diameter(F.n) by Def4;
    hence thesis by MEASURE5:13;
  end;
  hence thesis by SUPINF_2:39;
end;
