It is stated by Douglas Bridges in Constructive mathematics: a foundation for computable analysis that the following property, which I will call the zero product property:

If $x,y \in \mathbb{R}$ and $xy = 0$, then $x = 0$ or $y = 0$.

is equivalent to the Lesser Limited Principle of Omniscience (LLPO):

For each binary sequence $(a_n)$ with at most one term equal to 1, either $a_{2n} = 0$ for all $n$ or else $a_{2n+1} = 0$ for all $n$.

The LLOP is non-algorithmic and a trivial consequence of the Law of Excluded Middle (LEM); as such it is often rejected by constructive mathematicians. In light of this revelation how do construcivists operate, in general situations, without use of the zero product property?

Edit: The following example is referenced in some of the answers below but has been removed from the original post because it is receiving more attention than the fundamental question.

$$x^2-4 = 0 \implies (x-2)(x+2) = 0 \implies x-2 = 0 \ \text{or} \ x+2 = 0 \implies x = 2 \ \text{or} \ x = -2.$$

I realize the example given is fairly easy to reconcile without the zero product property. I'm looking for answers that give more general techniques employed to operate with out the zero product property. I've gathered that typically one must add extra assumptions (e.g. apartness of x and y.)

9more comments