
theorem ACG:
  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 AG2; then
    log (c,d) > 1 by A2,XXREAL_0:2;
    hence thesis by AG2;
  end;
