
theorem PXR1:
  for r be Real st 0 < r holds
  ex c be non empty positive-yielding XFinSequence of REAL st
  for x be Nat holds (polynom(c)).x = r
  proof
    let r be Real;
    assume AS: 0 < r;
    r is Element of REAL by XREAL_0:def 1;then
    reconsider z= <% r %> as XFinSequence of REAL;
    now let x be Real;
      assume x in rng z; then
      x in {r} by AFINSQ_1:33;
      hence 0 < x by AS,TARSKI:def 1;
    end;
    then
    A1: z is positive-yielding;
    reconsider z as non empty
    positive-yielding XFinSequence of REAL by A1;
    take z;
    len z = 1 by AFINSQ_1:34;
    then consider a be Real such that
    A2: a = z.0 &
    for x be Nat holds (seq_p(z)).x = a by ASYMPT_2:29;
    thus thesis by A2;
  end;
