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 Th7:
  inf Svc(M,union rng Sets) <= SUM Volume(M,Cvr)
proof
  set Q = SUM(vol(M,On Cvr));
  for x being ExtReal st x in rng Ser vol(M,On Cvr) ex y being
  ExtReal st y in rng Ser Volume(M,Cvr) & x <= y
  proof
    let x be ExtReal;
    assume x in rng Ser vol(M,On Cvr);
    then consider n being Element of NAT such that
A1: x = Ser(vol(M,On Cvr)).n by FUNCT_2:113;
    consider m being Nat such that
A2: for Sets being SetSequence of X holds for G be Covering of Sets,F
    holds (Partial_Sums vol(M,On G)).n <= (Partial_Sums Volume(M,G)).m by Th6;
    take Ser(Volume(M,Cvr)).m;
A3: for Sets being SetSequence of X, G be Covering of Sets,F holds Ser(vol
    (M,On G)).n <= Ser (Volume(M,G)).m
    proof
      let Sets be SetSequence of X;
      let G be Covering of Sets,F;
      (Partial_Sums vol(M,On G)).n <= (Partial_Sums Volume(M,G)).m by A2;
      then Ser(vol(M,On G)).n <= (Partial_Sums Volume(M,G)).m by Th1;
      hence Ser(vol(M,On G)).n <= Ser(Volume(M,G)).m by Th1;
    end;
    m in NAT by ORDINAL1:def 12;
    hence thesis by A1,A3,FUNCT_2:4;
  end;
  then
A4: SUM vol(M,On Cvr) <= SUM Volume(M,Cvr) by XXREAL_2:63;
  Q in Svc(M,union rng Sets) by Def7;
  then inf Svc(M,union rng Sets) <= Q by XXREAL_2:3;
  hence thesis by A4,XXREAL_0:2;
end;
