Prove that zero multiplied by zero is equal to zero. This is my proof:

So,  $0\cdot0=0$

And we know that

$a-a=0$

By substitution,

We have $(a-a)(a-a)=0$

Then by simplifying, $a^2-a^2+a^2-a^2=0$

and the we have $0-0=0$,

Therefore, $0=0$.

I am not sure about my answer. Will you please show me another way of proving it or some way to improve my answer? Thank you!
$0 \cdot 0 = 0$ because $a \cdot 0 = 0$, which is an axiom of multiplication in Peano arithmetic.
i will try the proof with "Reductio ad absurdum": 0x0&lt;>0

(a) If 0x0&lt;>0 then (a.1) 0x0>0 or (a.2) 0x0&lt;0.

   For example - a.1:
      0x0&gt;0 =&gt; multiply by 1 both sides, we have the following equation 
                (3) 1*0*0&gt;1*0            

    And for the same logic to a.2, we have the following:
                (4) 1*0*0&lt;1*0  

      For (3) and (4), we have that it's NOT possible that the same numbers:                   
      1*0*0  and 1*0 are less and greater simultaneously. Therefore, The only way is that the numbers are equal. For that, We Can conclude that 0*0=0.

 I don't pretty sure for the details of applying "Reductio ad absurdum" but I believe that could be a possible way to prove it.