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
Suppose . Multiplying both sides by we obtain:
Thus we have
For the converse, suppose . Subtracting from both sides, we obtain: . Now , since , thus we must have . But this means .
Thus we have .
Therefore, if , .
QED