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 Th10:
  C_Meas M is nonnegative
proof
  for r being ExtReal st r in rng (C_Meas M) holds 0 <= r
  proof
    let r be ExtReal;
    assume r in rng (C_Meas M);
    then consider A being object such that
A1: A in bool X and
A2: (C_Meas M).A = r by FUNCT_2:11;
    reconsider A as Subset of X by A1;
    now
      let p be ExtReal;
      assume p in Svc(M,A);
      then ex G be Covering of A,F st p = SUM vol(M,G) by Def7;
      hence 0 <= p by Th4,MEASURE6:2;
    end;
    then 0 is LowerBound of Svc(M,A) by XXREAL_2:def 2;
    then 0 <= inf Svc(M,A) by XXREAL_2:def 4;
    hence 0 <= r by A2,Def8;
  end;
  then for r being ExtReal holds r in rng C_Meas M implies 0 <= r;
  then rng C_Meas M is nonnegative;
  hence C_Meas M is nonnegative;
end;
