This is trivial. But I could not work out a formal prove for this (using the axioms of addition).

Seems like we have to rely of definition rather than a formal prove.

One approach:

Let us for the moment assume

A=B => B-A = 0 Let is call it (1)

Then

A+C = A + C + B - A = B + C

But to prove (1) above we somewhere have to rely on A=B => A+C = B+C

Any help please?

Thanks