((x / y) * y) = x is not a true statement about real numbers, regardless of if division by zero is defined as an error or as being equal to 0. The true statement is ( y != 0 -> ((x / y) * y) = x ), and this holds no matter how you define division by zero.
More generally, in normal math, every theorem that involves division by some value Y which could be zero must have as hypothesis that Y is not equal to zero. All these theorems still work in systems like Lean where division by zero is defined to be zero, so nothing is lost.
More generally, in normal math, every theorem that involves division by some value Y which could be zero must have as hypothesis that Y is not equal to zero. All these theorems still work in systems like Lean where division by zero is defined to be zero, so nothing is lost.