reserve r,t for Real;
reserve i for Integer;
reserve k,n for Nat;
reserve p for Polynomial of F_Real;
reserve e for Element of F_Real;
reserve L for non empty ZeroStr;
reserve z,z0,z1,z2 for Element of L;

theorem Th51:
  for p being monic INT -valued Polynomial of F_Real
  for e being rational Element of F_Real st e is_a_root_of p
  holds e is integer
  proof
    let p be monic INT -valued Polynomial of F_Real;
    let e be rational Element of F_Real;
    assume
A1: e is_a_root_of p;
    set k = numerator(e);
    set n = denominator(e);
A2: e = k/n by RAT_1:15;
A3: k,n are_coprime by WSIERP_1:22;
    p is monic;
    then n = 1 or n = -1 by A1,A2,A3,Th50,INT_2:13;
    hence thesis by A2;
  end;
