MATH.6:5.3 - A separate answer for each input and one answer for all inputs

Let X=Y={0,1} and let R(x,y) mean x≠y. For every x there is a y satisfying R: choose y=1-x. The proposed stronger conclusion is that one y works for every x.

To defeat that conclusion, take any proposed y and choose x=y. Then R fails. This gives the required argument for both possible choices of y. It does not replace the premise’s input-dependent choice with a uniform one.

The useful result can instead be a function h(x)=1-x, satisfying R(x,h(x)) for every x. MATH.4 supplies inductive witness construction when a comparable task has finite inductively formed inputs and suitable base and constructor clauses. The countermodel identifies which input dependence the requested result must retain.