
theorem ABA:
  for a,b be non weightless positive Real holds
  log (a,b) = 1/log (b,a)
  proof
    let a,b be non weightless positive Real;
    A1: a > 0 & b > 0 & a <> 1 & b <> 1 by TA1;
    a to_power log (a,b) = b & b to_power log (b,a) = a; then
    a = a to_power (log (a,b)*log (b,a)) by POWER:33; then
    (log (a,b))*(log (b,a)) = log (a,a)
    .= 1 by A1,POWER:52;
    hence thesis by XCMPLX_1:73;
  end;
