Then what do youmean"I want to prove" it? You can't prove something that isn't true!

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 thedefinitionof "a< b". How is that defined in your course?

One thing that it is important to realize- definitions in mathematics areworkingdefinitions- you use the precise words of definitions in proofs.