
theorem
  for F being sequence of bool REAL holds
  for G being Open_Interval_Covering of F
  holds inf Svc2(union rng F) <= SUM(vol(G))
proof
  let F be sequence of bool REAL;
  let G be Open_Interval_Covering of F;
  consider H being sequence of [:NAT,NAT:] such that
A1: H is one-to-one and
  dom H = NAT and
A2: rng H = [:NAT,NAT:] by MEASURE6:1;
  set GG = On(G,H);

A3: for x being ExtReal st x in rng Ser(GG vol) ex y being ExtReal
 st y in rng Ser(vol(G)) & x <= y
  proof
    let x be ExtReal;
    assume x in rng Ser(GG vol);
    then consider n being object such that
A4: n in dom Ser(GG vol) and
A5: x = Ser(GG vol).n by FUNCT_1:def 3;
    reconsider n as Element of NAT by A4;
    consider m being Element of NAT such that
A6: for F being sequence of bool REAL holds
     for G be Open_Interval_Covering of F holds
       Ser((On(G,H)) vol).n <= Ser(vol(G)).m by A1,A2,Th34;
    take Ser(vol(G)).m;
    dom Ser(vol(G)) = NAT by FUNCT_2:def 1;
    hence thesis by A5,A6,FUNCT_1:def 3;
  end;
  reconsider GG as Open_Interval_Covering of union rng F by A2,Th31;
  set Q = vol(GG);
  Q in Svc2(union rng F) by Def7; then
A7: inf Svc2(union rng F) <= Q by XXREAL_2:3;
  SUM(GG vol) <= SUM(vol G) by A3,XXREAL_2:63; then
  vol(GG) <= SUM(vol(G)) by MEASURE7:def 6;
  hence inf Svc2(union rng F) <= SUM(vol(G)) by A7,XXREAL_0:2;
end;
