
theorem Lemacik:
  for a,b,c being Real st a <= b & b > 0 & c >= 0
     holds a / b <= (a + c) / (b + c)
  proof
    let a,b,c be Real;
    assume
A1: a <= b & b > 0 & c >= 0; then
    a * c <= b * c by XREAL_1:64; then
    a * b + a * c <= a * b + b * c by XREAL_1:6; then
    a * (b + c) <= b * (a + c);
    hence thesis by XREAL_1:102,A1;
  end;
