
theorem
  for a,b,c being Real st a <= b & b < c holds a < (b+c)/2
  proof
    let a,b,c be Real;
    assume that
A1: a <= b and
A2: b < c;
A3: a + b < b + c by A1,A2,XREAL_1:8;
    a + a <= a + b by A1,XREAL_1:7;
    then 2 * a < b + c by A3,XXREAL_0:2;
    then 2 * a / 2 < (b + c) / 2 by XREAL_1:74;
    hence thesis;
  end;
