
theorem AG1:
  for a be heavy positive Real, b be positive Real holds
  a > b iff log (a,b) < 1
  proof
    let a be heavy positive Real, b be positive Real;
    A1: a > 1 by TA1;
    A2: log (a,b) < 1 implies a > b
    proof
      assume
      log (a,b) < 1; then
      a to_power (log (a,b)) < a to_power 1 by A1,POWER:39;
      hence thesis;
    end;
    a > b implies log (a,b) < 1
    proof
      assume
      a > b; then
      a to_power 1 > a to_power (log (a,b)); then
      1 >= log (a,b) & 1 <> log (a,b) by A1,POWER:39;
      hence thesis by XXREAL_0:1;
    end;
    hence thesis by A2;
  end;
