
theorem ABO:
  for a be non weightless positive Real, b be positive Real holds
  log (a,b) = -log (1/a,b) & log (1/a,b) = log (a,1/b) &
  log (a,b) = -log (a,1/b) & log (a,b) = log (1/a,1/b)
  proof
    let a be non weightless positive Real, b be positive Real;
    A1: a <> 1 by TA1;
    reconsider x = 1/a as positive Real;
    A3: 1/a <> 1/1;
    A5: x to_power (-1) = a to_power (-(-1)) by POWER:32;
    A6: log (x,b) = log (x,a) * log (a,b) by A3,POWER:56
    .= (-1)*(1)*log (a,b) by A5;
    A7: (1/b) to_power 1 = b to_power (-1) by POWER:32; then
    A8: log (a,1/b) = (-1)*log(a,b) by A1,POWER:55;
    log (x,1/b) = (-1)*log (x,b) by A3,A7,POWER:55;
    hence thesis by A6,A8;
  end;
