In group ring and field theory we often assume that we can add the same thing to both sides of an equation or multiply both sides of an equation by the same element.

How do we formally justify these operations or are they axioms?

That is, what is the rigorous and formal proof of the following:

1. If a = b then a+c = b + c and c + a = c + b

2. If a = b then ca = cb and ac = bc

Would appreciate help in this matter.

Peter