Proving Theorems with Axioms

Apologies for the lack of a better way to state the problem I'm having right now.

I'm trying to prove 2 theorems given by my professor:

1. ∀*a*∈ ℝ (Real numbers),

2. If a, b ∈ (real numbers) and a =/= 0 and b =/= 0, then a*b =/=. If a, b ∈ (real numbers) and a*b = 0 and a=/= 0, then b = 0

For the first one, I understand that I need to use the axiom which states: 1*a = a, but I don't know how to continue.

The second one, honestly, confuses me and I don't know where to start.