I need help with this proof also
x = y <=> xz = yz
use only addition and multiplication for proof
x, y and z are natural numbers
thanks.
I do not see a problem here, you just want to go from the right to the left and then from the left to the right. Only basic algebraic manipulation is required.
Proof
Let $\displaystyle x,y,z \in \mathbb{N}$
Suppose $\displaystyle x = y$. Multiplying both sides by $\displaystyle z$ we obtain: $\displaystyle xz = yz$
Thus we have $\displaystyle x = y \implies xz = yz$
For the converse, suppose $\displaystyle xz = yz$. Subtracting $\displaystyle yz$ from both sides, we obtain: $\displaystyle z(x - y) = 0$. Now $\displaystyle z \ne 0$, since $\displaystyle z \in \mathbb{N}$, thus we must have $\displaystyle x - y = 0$. But this means $\displaystyle x = y$.
Thus we have $\displaystyle xz = yz \implies x = y$.
Therefore, if $\displaystyle x,y,z \in \mathbb{N}$, $\displaystyle x = y \Longleftrightarrow xz = yz$.
QED