projective regular function chart compatibility

The intersection of the two charts with $D_{+}(HH\mathrm{\prime })\cap X$ contains the given point, so it is a nonempty open subset of the irreducible variety $X$. If the two evaluations agree on that overlap, the homogeneous cross difference $GH\mathrm{\prime }-G\mathrm{\prime }H$ vanishes there. Its zero locus is closed, and every nonempty open subset of an irreducible space is dense; hence the cross difference vanishes on all of $X$ and is zero in $S(X)$. Conversely, if $GH\mathrm{\prime }=G\mathrm{\prime }H$ in $S(X)$, evaluation and division by the nonzero product $HH\mathrm{\prime }$ show that the fractions agree throughout their common domain. Equivalently, this is equality of the two fractions in $\mathrm{Frac}(S(X))$.

Dehomogenization in either chart turns this into the same equality of ordinary affine fractions.

Therefore the affine descriptions agree exactly as stated.