Also a proof by contradiction:
Let's assume you are correct with you definition of 0 * 1 = 1, and we've already proved that the algebra version of your pattern is correct.
Now let's try to do 1 * 1.
Your pattern states this should be: 1 + (0 * 1) + 0
Hey! That 0 * 1 looks familiar, we have the answer for it as well, so let's substitute: 1 + 1 + 0
Now looks what you've done, 1 * 1 = 2
But we all know 1 * 1 = 1, therefore contradiction, so 0 * 1 cannot equal 1.
Let's assume you are correct with you definition of 0 * 1 = 1, and we've already proved that the algebra version of your pattern is correct.
Now let's try to do 1 * 1.
Your pattern states this should be: 1 + (0 * 1) + 0
Hey! That 0 * 1 looks familiar, we have the answer for it as well, so let's substitute: 1 + 1 + 0
Now looks what you've done, 1 * 1 = 2
But we all know 1 * 1 = 1, therefore contradiction, so 0 * 1 cannot equal 1.
