reserve X for set;

theorem Th21:
  for S being SigmaField of X, N,F being sequence of S holds (
F.0 = N.0 & for n being Nat holds F.(n+1) = N.(n+1) \ N.n & N.n c= N
  .(n+1) ) implies F is Sep_Sequence of S
proof
  let S be SigmaField of X, N,F be sequence of S;
  assume
A1: F.0 = N.0 & for n being Nat holds F.(n+1) = N.(n+1) \ N.n
  & N.n c= N. (n+1);
  for n,m being object st n <> m holds F.n misses F.m
  proof
    let n,m be object;
    assume
A2: n <> m;
    per cases;
    suppose
      n in dom F & m in dom F;
      then reconsider n9=n,m9=m as Element of NAT;
A3:   m9 < n9 implies F.m misses F.n by A1,Th19;
      n9 < m9 implies F.n misses F.m by A1,Th19;
      hence thesis by A2,A3,XXREAL_0:1;
    end;
    suppose
      not (n in dom F & m in dom F);
      then F.n = {} or F.m = {} by FUNCT_1:def 2;
      hence thesis;
    end;
  end;
  hence thesis by PROB_2:def 2;
end;
