
theorem Th24:
  for f be eventually-non-zero Real_Sequence holds
  for n be Nat holds f ^\ n is eventually-non-zero
  proof
    let f be eventually-non-zero Real_Sequence, n be Nat;
    set g = f ^\ n;
    let k be Nat;
    consider N be Nat such that
A1: k + n <= N & f.N <> 0 by ENZ;
    k <= N - n by A1,XREAL_1:19; then
    N - n in NAT by INT_1:3; then
    reconsider Nn = N - n as Nat;
    take Nn;
    g.Nn = f.(Nn+n) by NAT_1:def 3
        .= f.N;
    hence thesis by A1,XREAL_1:19;
  end;
