I don't know what you mean by "A is equal to B". In general, the hypothesis and conclusion of a proof (A and B) are statements or combinations of statements and it doesn't make sense to say "equal".I don't know how to do this kind of proofs.
I mean, when trying to prove
should I try to work on the left side of the implication until A is equal to B?
I have no idea what you are trying to do here.Am I allowed to take on what I have on the left side and make a substitution of that on the right side?
i.e., is this allowed?:
If you want to prove the "a< b" implis "a+ x= b for x> 0" then somewhere you are going to have to use the definition of "a< b". How is that defined in your course?
One thing that it is important to realize- definitions in mathematics are working definitions- you use the precise words of definitions in proofs.