In 1975 they did not prove Hawking radiation. That can only be done by experiment or direct observations, and there are none as yet. Neither have they disproved it in 2018. It is a hypothesis so far, no more.
I try to answer your question and for that I have to start with an analogy.
When I design and oscillator for an integrated circuit then I use there transistors capacitors, resistors and coils. With them I can extremely successfully create an oscillator with exact the required frequency behaviour. To do that I use equations. I use formulas with ejω components or in other words mathematics with imaginary, complex, values. They are very accurate. We have models for the transistors and other components and put all together in the simulators. Then the outcome is close to the real silicon. Why do I tell this with your question?
In the equations we have complex values but our electrons in our silicon circuit never go out of real space in an imaginary world. The behaviour of oscillation can be described with difficult real number equations of with much more elegant and understandable complex numbers. In the calculations for gravity Einstein came up with a calculation method for gravitational processes that makes use of curved space. Contrary to the situation in my silicon example where nobody expects that the electrons go in the complex number system space, we expect in the gravity calculations that space curves the same way that the equations describe. The calculations of space curvature is done in a model space, where deductions for gravity processes are, maybe easier, described. The problem is that it is assumed that space in reality, curves, wrinkles and waves according to the calculation model.
The Hawking radiation is a result of deep analysis of the consequences of curved space time. The reality in extreme situations is often a little bit different to the model. The more observations we have the more we will notice that these extreme corners of the model might be so different from reality that we need to conclude that the assumed processes like the Hawking radiation disappear.
In 1975 they did not prove Hawking radiation. That can only be done by experiment or direct observations, and there are none as yet. Neither have they disproved it in 2018. It is a hypothesis so far, no more.
-Did a black hole consist of a physical singularity or it is just an artificial mathematical flaw that can be avoided by developing a better theory?
-NASA Has Found Some of the Biggest and Oldest Black Holes We've Ever Seen. From the early moments of our Universe The discovery could change our understanding of how black holes came together in the earliest moments of the Universe, because these signals are coming from galaxies that formed when it was only 1.4 billion years old . It is hard to understand how a black hole became so big, so long ago. So why is that surprising? Black holes are like humans. They start out small and they grow over time. This particular black hole existed when the universe was only about 5% of its current age, and was located so far away that the light signaling its existence is only now arriving on Earth, after traveling for about thirteen billion years.
-Is a black hole well identified? Is it has an event horizon or an apparent horizon?
-The hypothetical Hawking radiation has never been observed, in order to be accepted as a scientific theory.
- How could we built speculation upon speculation?!
Inflation, singularity, black hole, dark matter/energy etc., ad nauseum, are fictions created by unlimited extension of idealized mathematics and are the results of a defective concept of gravity. Please see:
Article THE CONCEPTUAL DEFECT OF THE LAW OF UNIVERSAL GRAVITATION OR...