scheme :: HILB10_3:sch 3
IntersectionDiophantine{ F₁() -> Nat, P₁[ XFinSequence], P₂[ XFinSequence] } :

{ p where p is F₁() -element XFinSequence of NAT : ( P₁[p] & P₂[p] ) } is diophantine Subset of (F₁() -xtuples_of NAT)

provided

A1: { p where p is F₁() -element XFinSequence of NAT : P₁[p] } is diophantine Subset of (F₁() -xtuples_of NAT) and
A2: { p where p is F₁() -element XFinSequence of NAT : P₂[p] } is diophantine Subset of (F₁() -xtuples_of NAT)