reserve X for set;

theorem
  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.(meet rng F) = M.(F.0) -
  sup(rng (M*G))
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 = {} & for n being Nat holds G.(n+1) = F.0 \ F.n & F.
  (n+1) c= F .n;
  for n being Nat holds G.n c= G.(n+1) by A2,MEASURE2:13;
  then M.(union rng G) = sup(rng (M*G)) by MEASURE2:23;
  hence thesis by A1,A2,Th6;
end;
