
theorem TPOWSUCC:
  for k,n be Nat st 0 < n holds
    n*(seq_n^(k)).n = ((seq_n^(k+1)).n)
  proof
    let k,n be Nat;
    ZZ: k in NAT & n in NAT by ORDINAL1:def 12;
    assume AS: 0 < n;
    (seq_n^(k+1)).n = n to_power (k+1) by ZZ,ASYMPT_1:def 3,AS
    .= (n to_power k)*(n to_power 1) by POWER:27,AS
    .= (n to_power k)* n by POWER:25;
    hence thesis by AS,ZZ,ASYMPT_1:def 3;
  end;
