 reserve n,k for Nat;
 reserve L for comRing;
 reserve R for domRing;
 reserve x0 for positive Real;

theorem Th15:
  for R be comRing
  for f be Element of the carrier of Polynom-Ring R holds
  for i be Nat st i >= 1 & i is_at_least_length_of f & f.(i-1) <> 0.R holds
    len f = i
  proof
    let R be comRing;
    let f be Element of the carrier of Polynom-Ring R;
    for i be Nat st i >= 1 & i is_at_least_length_of f & f.(i-1) <> 0.R
    holds len f = i
    proof
      let i be Nat;
      assume
A1:   i >= 1 & i is_at_least_length_of f & f.(i-1) <> 0.R; then
A2:   len f <= i by ALGSEQ_1:def 3;
   for n,m be Nat st n -1 < m <= n holds n = m
      proof
        let n,m be Nat;
        assume
A4:     n -1 < m <= n; then
        n-1 +1 < m + 1 by XREAL_1:8; then
        n <= m by NAT_1:13;
        hence thesis by A4,XXREAL_0:1;
      end;
      hence thesis by A2,A1,ALGSEQ_1:8;
    end;
    hence thesis;
  end;
