The requirement that black holes lose energy into the wider universe, and therefore can "evaporate", and the radiated spectrum are both a result of analysing black hole thermal equilibrium combined with extreme redshifting effects very close to the event horizon, with some consideration of quantum entanglement effects. A pair of virtual waves/particles arises just outside the event horizon due to ordinary quantum effects. Very close to the event horizon, these always manifest as a pair of photons. It may happen that one of these photons passes beyond the event horizon, while the other escapes into the wider universe ("to infinity"). A close analysis shows that the exponential red-shifting effect of extreme gravity very close to the event horizon almost tears the escaping photon apart, and in addition very slightly amplifies it. The amplification gives rise to a "partner wave", which carries negative energy and passes through the event horizon, where it remains trapped, reducing the total energy of the black hole. The escaping photon adds an equal amount of positive energy to the wider universe outside the black hole. In this way, no matter or energy ever actually leaves the black hole itself. A conservation law exists for the partner wave, which in theory shows that the emissions comprise an exact black body spectrum, bearing no information about the interior conditions.