
theorem
  for a be Real st 1 < a holds seq_a^(a,1,0) is increasing
  proof
    let a be Real;
    assume AS: 1 < a;
    C1:for n be Element of NAT holds
    seq_a^(a,1,0).n < seq_a^(a,1,0).(n+1)
    proof
      let n be Element of NAT;
      L2:(seq_a^(a,1,0)).n =a to_power (1*n+0) by ASYMPT_1:def 1
      .= a to_power n;
      (seq_a^(a,1,0)).(n+1) =a to_power (1*(n+1)+0) by ASYMPT_1:def 1
      .= a to_power (n+1);
      hence thesis by L2,LC5aa,AS;
    end;
    reconsider S = seq_a^(a,1,0) as Real_Sequence;
    for n be Nat holds S.n < S.(n+1)
    proof
      let n be Nat;
      reconsider n as Element of NAT by ORDINAL1:def 12;
      seq_a^(a,1,0).n < seq_a^(a,1,0).(n+1) by C1;
      hence thesis;
    end;
    hence thesis by SEQM_3:def 6;
  end;
