reserve X for set;

theorem Th10:
  for S being SigmaField of X holds for N, G, F being sequence of S holds
  G.0 = N.0 & (for n being Nat holds G.(n+1) = N.(n+1) \/
  G.n) & F.0 = N.0 & (for n being Nat holds F.(n+1) = N.(n+1) \ G.n)
  implies for n,m being Nat st n < m holds F.n misses F.m
proof
  let S be SigmaField of X;
  let N, G, F be sequence of S;
  assume that
A1: ( G.0 = N.0 & for n being Nat holds G.(n+1) = N.(n+1) \/
  G.n )& F.0 = N.0 and
A2: for n being Nat holds F.(n+1) = N.(n+1) \ G.n;
  let n,m be Nat;
  assume
A3: n < m;
  then 0 <> m by NAT_1:2;
  then consider k being Nat such that
A4: m = k + 1 by NAT_1:6;
  reconsider k as Element of NAT by ORDINAL1:def 12;
  F.(k+1) = N.(k+1) \ G.k by A2;
  then
A5: G.k misses F.(k+1) by XBOOLE_1:79;
  n <= k by A3,A4,NAT_1:13;
  hence F.n /\ F.m = (F.n /\ G.k) /\ F.(k+1) by A1,A2,A4,Th7,XBOOLE_1:28
    .= F.n /\ (G.k /\ F.(k+1)) by XBOOLE_1:16
    .= F.n /\ {} by A5
    .= {};
end;
