reserve X for set;

theorem
  for S being SigmaField of X, M being sigma_Measure of S, F being
  sequence of S, A being Element of S st meet rng F c= A & (for n being
  Element of NAT holds A c= F.n) holds M.A = M.(meet rng F)
proof
  let S be SigmaField of X;
  let M be sigma_Measure of S;
  let F be sequence of S;
  let A be Element of S;
  assume that
A1: meet rng F c= A and
A2: for n being Element of NAT holds A c= F.n;
  A c= meet rng F
  proof
    let x be object;
    assume
A3: x in A;
    for Y being set st Y in rng F holds x in Y
    proof
      let Y be set;
A4:   dom F = NAT by FUNCT_2:def 1;
      assume Y in rng F;
      then ex n being object st n in NAT & Y = F.n by A4,FUNCT_1:def 3;
      then A c= Y by A2;
      hence thesis by A3;
    end;
    hence thesis by SETFAM_1:def 1;
  end;
  then
A5: M.(A) <= M.(meet rng F) by MEASURE1:31;
  M.(meet rng F) <= M.(A) by A1,MEASURE1:31;
  hence thesis by A5,XXREAL_0:1;
end;
