I'm trying to prove that $a\cdot b=b\cdot a$ when $a$ and $b$ are two natural numbers.
In the rest of this question I'm using $a'$ for the successor of $a$.
Addition is defined as:
Multiplication is defined as:
- $a\cdot 0=0$
- $a\cdot b'=a+ab$
I already proved commutativity and associativity for addition. I also proved that $a\cdot 1=1\cdot a=a$.
I tried with induction on $b$. I can easily show that $a\cdot 0=0\cdot a$. Then I suppose $a\cdot b=b\cdot a$ and try to show that $a\cdot b'=b'\cdot a$.
Here I can no longer go on. The main problem is I can't use distributivity laws since I haven't proved them yet. I hope to do that immediately after this problem is fixed. Also, $b'\cdot a$ is problematic because $b'$ is at the left.