Section PHP_rel.
Variable (U V : Type) (R : U -> V -> Prop) (l : list U) (m : list V)
(HR : forall x, In x l -> exists y, In y m /\ R x y).
Let image_R_l : exists Rl, incl Rl m /\ Forall2 R l Rl.
Hypothesis (Hlm : length m < length l).
(* ** The finite relational PHP *)

Theorem PHP_rel : exists a x b y c v, l = a++x::b++y::c
/\ In v m /\ R x v /\ R y v.