reserve X for set;

theorem Th9:
  for S being SigmaField of X, M being sigma_Measure of S, G,F
being sequence of S st (M.(F.0) <+infty & G.0 = {} & for n being Nat
   holds G.(n+1) = F.0 \ F.n & F.(n+1) c= F.n )
holds M.(F.0) in REAL & inf(rng (M*F)) in REAL & sup(rng (M*G)) in REAL
proof
  let S be SigmaField of X, M be sigma_Measure of S, G,F be sequence of S;
  assume that
A1: M.(F.0) <+infty and
A2: G.0 = {} and
A3: for n being Nat holds G.(n+1) = F.0 \ F.n & F.(n+1) c= F .n;
  reconsider P = {} as Element of S by PROB_1:4;
A4: 0 in REAL by XREAL_0:def 1;
  M.P <= M.(F.0) by MEASURE1:31,XBOOLE_1:2;
  then 0. <= M.(F.0) by VALUED_0:def 19;
  hence
A5: M.(F.0) in REAL by A1,XXREAL_0:46,A4;
  for x being ExtReal st x in rng(M*G) holds x <= M.(F.0)
  proof
    let x be ExtReal;
A6: dom (M*G) = NAT by FUNCT_2:def 1;
    assume x in rng(M*G);
    then consider n being object such that
A7: n in NAT and
A8: (M*G).n = x by A6,FUNCT_1:def 3;
    reconsider n as Element of NAT by A7;
A9: x = M.(G.n) by A6,A8,FUNCT_1:12;
A10: (ex k being Nat st n = k + 1) implies x <= M.(F.0)
    proof
      given k being Nat such that
A11:  n = k + 1;
      reconsider k as Element of NAT by ORDINAL1:def 12;
      G.n = F.0 \ F.k by A3,A11;
      hence thesis by A9,MEASURE1:31,XBOOLE_1:36;
    end;
    n = 0 implies x <= M.(F.0) by A2,A9,MEASURE1:31,XBOOLE_1:2;
    hence thesis by A10,NAT_1:6;
  end;
  then M.(F.0) is UpperBound of rng(M*G) by XXREAL_2:def 1;
  then
A12: sup(rng(M*G)) <= M.(F.0) by XXREAL_2:def 3;
  for x being ExtReal st x in rng(M*F) holds 0.<= x
  proof
    let x be ExtReal;
A13: dom (M*F) = NAT by FUNCT_2:def 1;
A14: (M*F) is nonnegative by MEASURE2:1;
    assume x in rng(M*F);
    then ex n being object st n in NAT & (M*F).n = x by A13,FUNCT_1:def 3;
    hence thesis by A14,SUPINF_2:39;
  end;
  then 0. is LowerBound of rng(M*F) by XXREAL_2:def 2;
  then
A15: inf(rng(M*F)) >= In(0,REAL) by XXREAL_2:def 4;
  ex x being R_eal st x in rng(M*F) & x = M.(F.0)
  proof
    take (M*F).0;
    dom (M*F) = NAT by FUNCT_2:def 1;
    hence thesis by FUNCT_1:12,FUNCT_2:4;
  end;
  then inf(rng(M*F)) <= M.(F.0) by XXREAL_2:3;
  hence inf(rng(M*F)) in REAL by A5,A15,XXREAL_0:45;
  In(0,REAL) <= sup(rng(M*G))
  proof
    set x = (M*G).0;
    for x being R_eal st x in rng(M*G) holds 0.<= x
    proof
      let x be R_eal;
A16:  dom (M*G) = NAT by FUNCT_2:def 1;
A17:  (M*G) is nonnegative by MEASURE2:1;
      assume x in rng(M*G);
      then ex n being object st n in NAT & (M*G).n = x by A16,FUNCT_1:def 3;
      hence thesis by A17,SUPINF_2:39;
    end;
    then
A18: 0. <= x by FUNCT_2:4;
    x <= sup rng(M*G) by FUNCT_2:4,XXREAL_2:4;
    hence thesis by A18,XXREAL_0:2;
  end;
  hence thesis by A5,A12,XXREAL_0:45;
end;
