
theorem LemmaImp:
  for x, r being Real st 0 <= x <= 1 & r > -1 holds
    x * r + 1 > 0
  proof
    let x, r be Real;
    assume
AA: 0 <= x <= 1 & r > -1;
    per cases;
    suppose r < 0; then
      r <= r * x by XREAL_1:152,AA; then
      -1 < r * x by XXREAL_0:2,AA; then
      x * r + 1 > -1 + 1 by XREAL_1:8;
      hence thesis;
    end;
    suppose r >= 0;
      hence thesis by AA;
    end;
  end;
