
theorem TH3:
  for f be Function of NAT,REAL
  st for x be Nat holds f.x = 1/ (2 to_power x)
  holds f is negligible
  proof
    let f be Function of NAT,REAL;
    assume AS: for x be Nat holds f.x = 1/ (2 to_power x);
    let c be non empty positive-yielding XFinSequence of REAL;
    set k = len c;
    deffunc F(Nat) = 1*($1 to_power k);
    consider y be Real_Sequence such that
    P1: for x be Nat holds y.x = F(x) from SEQ_1:sch 1;
    consider N1 be Nat such that
    P2: for x be Nat st N1 <=x holds |.(seq_p(c)).x .| <= y.x
    by ASYMPT_2:43,P1;
    consider N2 be Nat such that
    P3: for x be Nat st N2 <= x
    holds 1/ (2 to_power x) < 1/ (x to_power k) by L2;
    set N=N1+N2;
    reconsider N as Element of NAT by ORDINAL1:def 12;
    take N;
    thus for x be Nat
    st N <=x holds |. f.x .| < 1/((polynom(c)).x)
    proof
      let x be Nat;
      assume D3:N <=x;
      K1: f.x = 1/ (2 to_power x) by AS;
      N2 <= N by NAT_1:12; then
      N2<= x by D3,XXREAL_0:2; then
      1/ (2 to_power x) < 1/ (x to_power k) by P3; then
      P4: |.f.x .| < 1/ (x to_power k) by K1,ABSVALUE:def 1;
      N1 <= N by NAT_1:12; then
      N1<= x by D3,XXREAL_0:2; then
      |.(polynom(c)).x .| <= y.x by P2; then
      (polynom(c)).x <= y.x by ABSVALUE:def 1; then
      (polynom(c)).x <= 1*(x to_power k) by P1; then
      1/ x to_power k <= 1/ ((polynom(c)).x) by XREAL_1:118,NLM3;
      hence |.f.x .| < 1/ ((polynom(c)).x) by P4,XXREAL_0:2;
    end;
  end;
