Hi i am a math student , in the first proof i think you started well but when you take lBl+1=k+1 it same as lBl=k , i think you have to prove for lBl=k+1, and i don't know what the successor function can you please explain ?
The second proof is very simple

It's helpful to your readers to say what you are trying to prove about it. Are you trying to prove that the Cartesian product exists? That for finite sets it has a particular cardinality? That it has the universal property that characterizes it in category theory?

You have to say what you are trying to prove. In fact one of the key "tricks" for doing proofs is to state clearly and precisely exactly what it is you're trying to prove; and exactly what you have to show in order to prove it.

When you state the problem in very clear and explicit terms, the solution often falls out directly.

So: What are you trying to prove? And what do you have to show in order to prove it?

Just to note, you're not proving the cartesian product, but the properties of cartesian products. Otherwise, you're claiming to prove a definition, which is logically impossible (?).

Let's assume the property is true for lBl=k
then we will prove for lBl=k+1
let B_{k+1}={b_{1},b_{2},b_{3},......b_{k+1} }
let B_{k}={b_{1},b_{2},...b_{k}} then B_{k+1}=B_{k}U{b_{k+1}} and lB_{k}l=k
now using the property Ax(BUC)=(AxB)U(AxC)
we have lAxB_{k+1}l =l(AxB_{k})U(Ax{b_{k+1}})l
=lAxB_{k}l+lAx{b_{k+1}}l since b_{k+1} is not in B_{k}
then the proof follows from the fact that its true for k , am i right ?