
theorem Th40:
  for nl be Liouville, z be Integer holds z + nl is liouville
  proof
    let nl be Liouville, z be Integer;
    set zl = z + nl;
    for n be non zero Nat ex p be Integer, q be Nat
    st q > 1 & 0 < |. zl - p/q .| < 1/q|^n
    proof
      let n be non zero Nat;
      nl is liouville; then
      consider p1 be Integer, q1 be Nat such that
A1:   q1 > 1 & 0 < |. nl - p1/q1 .| < 1/q1|^n;
      set p2 = z * q1 + p1;
      take p2, q2 = q1;
      thus q2 > 1 by A1;
      p2/q2 = (z * q1)/q1 + p1/q1 by XCMPLX_1:62
             .= z + p1/q1 by A1,XCMPLX_1:89;
      hence thesis by A1;
    end;
    hence thesis by Def2;
  end;
