reserve i,i1,i2,i3,i4,i5,j,r,a,b,x,y for Integer,
  d,e,k,n for Nat,
  fp,fk for FinSequence of INT,
  f,f1,f2 for FinSequence of REAL,
  p for Prime;

theorem Th3:
  len fp = 1 implies (Poly-INT fp) = INT --> fp.1
proof
  assume
A1: len fp = 1;
  for x being object st x in dom Poly-INT fp holds (Poly-INT fp).x = fp.1
  proof
    let x be object;
    assume x in dom Poly-INT fp;
    then reconsider x as Element of INT;
    consider fr being FinSequence of INT such that
A2: len fr = len fp and
A3: for d st d in dom fr holds fr.d = (fp.d) * x|^(d-'1) and
A4: (Poly-INT fp).x = Sum fr by Def1;
    1 in dom fr by A1,A2,FINSEQ_3:25;
    then
A5: fr.1 = (fp.1) * x|^(1-'1) by A3
      .= fp.1 * x|^0 by XREAL_1:232
      .= fp.1 * 1 by NEWTON:4;
    fr = <*fr.1*> by A1,A2,FINSEQ_1:40;
    hence thesis by A4,A5,RVSUM_1:73;
  end;
  then Poly-INT fp = dom Poly-INT fp --> fp.1 by FUNCOP_1:11;
  hence thesis by FUNCT_2:def 1;
end;
