
theorem TH4:
  for f,g be Function of NAT,REAL
  st f is negligible &
  for x be Nat holds |. g.x .| <= |. f.x .| holds
  g is negligible
  proof
    let f,g be Function of NAT,REAL;
    assume AS: f is negligible &
    for x be Nat holds |. g.x .| <= |.f.x.|;
    thus g is negligible
    proof
      let c be non empty positive-yielding XFinSequence of REAL;
      consider N be Nat such that
      D3: for x be Nat st N <=x holds |. f.x .| < 1/((polynom(c)).x) by AS;
      take N;
      let x be Nat;
      assume N <=x;
      then
D2:   |. f.x .| < 1/((polynom(c)).x) by D3;
      |. g.x .| <= |. f.x .| by AS;
      hence thesis by D2,XXREAL_0:2;
    end;
  end;
