 reserve R for finite Approximation_Space;
 reserve X,Y,Z for Subset of R;
 reserve kap for RIF of R;

theorem LemacikX:
  for a,b,c being Real st a <= b & b > 0 & c >= 0 & b > c
     holds a / b >= (a - c) / (b - c)
  proof
    let a,b,c be Real;
    assume
A1: a <= b & b > 0 & c >= 0 & b > c; then
    b - 0 > c; then
SS: b - c > 0 by XREAL_1:12;
    a * c <= b * c by A1,XREAL_1:64; then
    a * b - a * c >= a * b - b * c by XREAL_1:10; then
    a * (b - c) >= b * (a - c);
    hence thesis by XREAL_1:102,A1,SS;
  end;
