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For example, we could use alittle algebra and rearrange the terms so we have : 1 - 1 = 0, but that already assumes we believe in the correctness of algebratic operations, and the orrectness of arithmetic, and that anything minus an equal amount equals "zero".

One possible approach is to say: If we are going to be able to prove anything at all, we have to make a few assumptions, such as the location of a symbol on a page does not change its value, and a symbol is always numerically equivalent to itself. If we can't make those assumptions, then nothing else we write is going to manke any sense, so we might as well just ASSUME that A = A, for all A, assuming A is a constant. There's probably a logical proof of this in Principia Mathematica.