
theorem LM1:
  for r be Element of NAT holds
  for s be Real_Sequence
  st s = NAT --> r
  holds s is polynomially-abs-bounded
  proof
    let r be Element of NAT;
    let s1 be Real_Sequence;
    assume AS: s1 = NAT --> r;
    set s=|.s1.|;
    P1: s in Funcs(NAT, REAL) by FUNCT_2:8;
    now
      let n be Element of NAT;
      assume A2: n >= r;
      n = 0 or n > 0;
      then
A4:   (seq_n^ 1) . n = n to_power 1 by ASYMPT_1:def 3
      .=n;
      A5: s.n = |. s1.n .| by VALUED_1:18
      .= |. r .| by AS
      .= r by ABSVALUE:def 1;
      hence s.n <= 1*(seq_n^ 1).n by A2,A4;
      thus s.n >= 0 by A5;
    end;
    then
    s in Big_Oh(seq_n^ 1) by P1;
    hence thesis;
  end;
