Section PolyXY_Field.
Variables (F E : fieldType) (FtoE : {rmorphism F -> E}).
Local Notation pFtoE := (map_poly (GRing.RMorphism.apply FtoE)).

Lemma map_sub_annihilantP (p q : {poly F}) (x y : E) :
p != 0 -> q != 0 ->(p ^ FtoE).[x] = 0 -> (q ^ FtoE).[y] = 0 ->
(sub_annihilant p q ^ FtoE).[x - y] = 0.