
theorem Th8:
  for T being non empty RelStr, p being Element of T, k being
  Nat st T is filled holds U_FT(p,0) c= U_FT(p,k)
proof
  let T be non empty RelStr, p be Element of T, k be Nat;
  defpred P[Nat] means U_FT(p,0) c= U_FT(p,$1);
  assume
A1: T is filled;
A2: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A3: P[k];
    U_FT(p,k) c= U_FT(p,k+1) by A1,Th7;
    hence thesis by A3,XBOOLE_1:1;
  end;
A4: P[0];
  for i being Nat holds P[i] from NAT_1:sch 2(A4,A2);
  hence thesis;
end;
