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 Th12:
  A c= B implies (C_Meas M).A <= (C_Meas M).B
proof
  assume
A1: A c= B;
  now
    let r be object;
    assume r in Svc(M,B);
    then consider G be Covering of B,F such that
A2: SUM vol(M,G) = r by Def7;
    B c= union rng G by Def3;
    then A c= union rng G by A1;
    then reconsider G1 = G as Covering of A,F by Def3;
    SUM vol(M,G) = SUM vol(M,G1);
    hence r in Svc(M,A) by A2,Def7;
  end;
  then
A3: Svc(M,B) c= Svc(M,A);
  (C_Meas M).A = inf Svc(M,A) & (C_Meas M).B = inf Svc(M,B) by Def8;
  hence (C_Meas M).A <= (C_Meas M).B by A3,XXREAL_2:60;
end;
