
theorem
  for a,c be heavy positive Real, b,d be positive Real
  st log (a,b) <= log (c,d) & a <= b holds c <= d
  proof
    let a,c be heavy 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 AG1;
  end;
