reserve X for set;

theorem
  for S being SigmaField of X, M being sigma_Measure of S, F being
sequence of S st (for n being Nat holds F.(n+1) c= F.n) & M.(F.0
  ) <+infty holds M.(meet rng F) = inf(rng (M*F))
proof
  let S be SigmaField of X, M be sigma_Measure of S, F be sequence of S;
  assume that
A1: for n being Nat holds F.(n+1) c= F.n and
A2: M.(F.0) <+infty;
  consider G being sequence of S such that
A3: G.0 = {} & for n being Nat holds G.(n+1) = F.0 \ F.n by
MEASURE2:9;
A4: union rng G = F.0 \ meet rng F by A1,A3,Th4;
A5: M.(F.0 \ union rng G) = M.(meet rng F) by A1,A3,Th5;
A6: for A being Element of S st A = union rng G holds M.(meet rng F) = M.(F.
  0) - M.A
  proof
    let A be Element of S;
    assume
A7: A = union rng G;
    M.(F.0 \ meet rng F) <> +infty by A2,MEASURE1:31,XBOOLE_1:36;
    then M.A <+infty by A4,A7,XXREAL_0:4;
    hence thesis by A4,A5,A7,MEASURE1:32,XBOOLE_1:36;
  end;
  for n being Nat holds G.n c= G.(n+1) by A1,A3,MEASURE2:13;
  then M.(union rng G) = sup(rng (M*G)) by MEASURE2:23;
  then M.(meet rng F) = M.(F.0) - sup(rng (M*G)) by A6;
  hence thesis by A1,A2,A3,Th11;
end;
