
theorem Th8:
  for a,b being Real st a <= b holds a <= (a+b)/2 <= b
  proof
    let a,b be Real;
    assume
A1: a <= b;
    2 * a = a + a;
    then 2 * a <= a + b by A1,XREAL_1:7;
    then 2 * a / 2 <= (a + b)/2 by XREAL_1:72;
    hence a <= (a + b)/2;
    a+b <= b + b by A1,XREAL_1:7;
    then (a+b)/2 <= 2 * b /2 by XREAL_1:72;
    hence thesis;
  end;
