
theorem
  for a,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,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,ACG;
    hence thesis by XREAL_1:118;
  end;
