
theorem RNG0:
  for c,s be non empty positive-yielding XFinSequence of REAL
  ex d be non empty positive-yielding XFinSequence of REAL,
  N be Nat
  st for x be Nat st N<= x holds
  (polynom(c)).x * (polynom(s)).x <= (polynom(d)).x
  proof
    let c,s be non empty positive-yielding
    XFinSequence of REAL;
    consider a1 be Real, k1,N1 be Nat such that
    P1: 0 < a1 & 0 < k1
    & for x be Nat st N1<= x
    holds (polynom(c)).x <=a1* (x to_power k1) by LRNG1;
    consider a2 be Real, k2,N2 be Nat such that
    P2: 0 < a2 & 0 < k2
    & for x be Nat st N2<= x
    holds (polynom(s)).x <=a2* (x to_power k2) by LRNG1;
    consider d be non empty positive-yielding
    XFinSequence of REAL such that
    P3: for x be Nat
    holds (a1*a2)*(x to_power (k1+k2)) <=(polynom(d)).x by LRNG2;
    set N = N1+N2;
    take d,N;
    let x be Nat;
    assume P4:N <=x;
    N1<=N by NAT_1:12; then
    N1<=x by XXREAL_0:2,P4; then
    P5: (polynom(c)).x <=a1* (x to_power k1) by P1;
    N2<=N by NAT_1:12; then
    N2<=x by XXREAL_0:2,P4; then
    P6: (polynom(s)).x <=a2* (x to_power k2) by P2;
    P7: 0 < (polynom(c)).x by NLM3;
    P8: 0 < (polynom(s)).x by NLM3;
    P9: (polynom(c)).x * (polynom(s)).x
    <= (a1* (x to_power k1))*(a2* (x to_power k2))
    by XREAL_1:66,P5,P6,P7,P8;
    P10: (a1* (x to_power k1))*(a2* (x to_power k2))
    =(a1*a2)*(x to_power (k1+k2))
    proof
      per cases;
      suppose P11: x=0;
        P12: x to_power k1 = 0 by P1,P11,POWER:def 2;
        x to_power (k1+k2) =0 by P2,P11,POWER:def 2;
        hence thesis by P12;
      end;
      suppose x<>0; then
        (x to_power k1) * (x to_power k2)
        = x to_power (k1+k2) by POWER:27;
        hence thesis;
      end;
    end;
    (a1*a2)*(x to_power (k1+k2)) <= (polynom(d)).x by P3;
    hence (polynom(c)).x * (polynom(s)).x
    <=(polynom(d)).x by P9,P10,XXREAL_0:2;
  end;
