
theorem
  for a be heavy positive Real, c be light positive Real, b,d be positive Real
  st log (a,b) >= log (c,d) & a > b holds c < d
  proof
    let a be heavy positive Real, c be light positive Real,
        b,d be positive Real;
    assume
    A2: log (a,b) >= log (c,d) & a > b; then
    log (a,b) < 1 by AG1; then
    log (c,d) < 1 by A2,XXREAL_0:2;
    hence thesis by AM1;
  end;
