reserve a,b,i,j,k,l,m,n for Nat;

theorem SN:
  for f be nonnegative-yielding real-valued FinSequence holds Sum f >= f.n :: POLYNOM5:4
  proof
    let f be nonnegative-yielding real-valued FinSequence;
    per cases;
    suppose not n in dom f;
      hence thesis by FUNCT_1:def 2;
    end;
    suppose
      n in dom f; then
      A0a: len f >= n >= 1 by FINSEQ_3:25; then
      reconsider k = n - 1 as Nat;
       k+1 > k+0 by XREAL_1:6; then
      len f > k by A0a,XXREAL_0:2; then
      A1: f|(k+1) = f|k^ <*f.(k+1)*> by FINSEQ_5:83;
      A2: Sum f = Sum (((f|k)^<*f.(k+1)*>)^(f/^n)) by A1,RFINSEQ:8
      .= Sum ((f|k)^<*f.(k+1)*>) + Sum (f/^n) by RVSUM_1:75
      .= Sum(f|k) + f.(k+1) + Sum (f/^n) by RVSUM_1:74;
      f.(k+1) + (Sum(f|k) + Sum (f/^n)) >= f.(k+1) + 0 by XREAL_1:6;
      hence thesis by A2;
    end;
  end;
