reserve X for set;

theorem
  for S being SigmaField of X, N,F being sequence of S holds (F.0 =
  {} & for n being Nat holds F.(n+1) = N.0 \ N.n & N.(n+1) c= N.n )
  implies rng F is non-decreasing N_Measure_fam of S
proof
  let S be SigmaField of X, N,F be sequence of S;
  assume F.0 = {} & for n being Nat holds F.(n+1) = N.0 \ N.n & N.
  (n+1) c= N. n;
  then
A1: for n being Nat holds F.n c= F.(n+1) by Th13;
  rng F c= S & rng F is N_Sub_set_fam of X by MEASURE1:23,RELAT_1:def 19;
  hence thesis by A1,Def1,Def2;
end;
