
theorem
  for a,c be light positive Real, b,d be positive Real holds
  log (a,b) >= log (c,d) & a < b implies c < d
  proof
    let a,c be light positive Real, b,d be positive Real;
    assume
    A3: log (a,b) >= log (c,d) & a < b;
    A4: log (a,b) = log (1/a,1/b) & log (c,d) = log (1/c,1/d) by ABO;
    1/a > 1/b by A3,XREAL_1:76; then
    1/c > 1/d by A3,A4,ACL;
    hence thesis by XREAL_1:118;
  end;
