
theorem NLM3:
  for c be non empty positive-yielding XFinSequence of REAL,
  x be Nat
  holds 0 < (polynom(c)).x
  proof
    defpred P[Nat] means
    for c be non empty positive-yielding XFinSequence of REAL
    st len c = $1 holds
    for x be Nat
    holds 0 < (polynom(c)).x;
    P0: P[0];
    P1: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume A1: P[k];
      let d be non empty positive-yielding XFinSequence of REAL;
      assume A2: len d = k+1;
      then
      consider a be Real,d1 be XFinSequence of REAL,
      y be Real_Sequence such that
      A3: len d1 = k & d1= d | k & a = d.k &
      d =d1^<% a %> & polynom(d) = polynom(d1) + y &
      for i be Nat holds
      y.i = a* (i to_power k) by ASYMPT_2:28;
      per cases;
      suppose X1:k = 0; then
        consider a be Real such that
        A4:a = d.0 &
        for x be Nat holds (polynom(d)).x = a by ASYMPT_2:29,A2;
        0 in Segm (0+1) by NAT_1:44;
        then d.0 in rng d by FUNCT_1:3,X1,A2;
        then
        A5:0 < d.0 by PARTFUN3:def 1;
        let x be Nat;
        reconsider n =x as Nat;
        thus 0 < (polynom(d)).x by A5,A4;
      end;
      suppose A7:k <> 0;
        now let r be Real;
          assume r in rng d1; then
          consider x be object such that
          P4: x in dom d1 & r = d1.x by FUNCT_1:def 3;
          P5: d1.x = d.x by P4,A3,FUNCT_1:47;
          x in dom d by P4,A3,RELAT_1:60,TARSKI:def 3;
          then
          d.x in rng d by FUNCT_1:3;
          hence 0 < r by PARTFUN3:def 1,P5,P4;
        end;
        then
        A8: d1 is positive-yielding;
        reconsider d1 as non empty positive-yielding XFinSequence of REAL
        by A7,A8,A3;
        let x be Nat;
        reconsider n=x as Nat;
        A9: 0 < (polynom(d1)).n by A1,A3;
        A10: (polynom(d)).n
        = (polynom(d1)).n + y.n by SEQ_1:7,A3;
        A11: y.n = a* (n to_power k) by A3;
        k < k+1 by NAT_1:13;
        then k in Segm (k+1) by NAT_1:44;then
        d.k in rng d by FUNCT_1:3,A2;
        then
        0 < d.k by PARTFUN3:def 1;
        hence 0 < (polynom(d)).x by A10,A9,A11,A3;
      end;
    end;
    P2: for k be Nat holds P[k] from NAT_1:sch 2(P0,P1);
    let c be non empty positive-yielding XFinSequence of REAL,
    x be Nat;
    len c is Nat;
    hence 0 < (polynom(c)).x by P2;
  end;
