Record partial_injection : Type := {
  pi_dom: Ensemble X;
  pi_func: forall x:X, In pi_dom x -> Y;
  pi_inj: forall (x1 x2:X) (i1:In pi_dom x1) (i2:In pi_dom x2),
    pi_func x1 i1 = pi_func x2 i2 -> x1 = x2
}.Record partial_injection : Type := {
  pi_dom: Ensemble X;
  pi_func: forall x:X, In pi_dom x -> Y;
  pi_inj: forall (x1 x2:X) (i1:In pi_dom x1) (i2:In pi_dom x2),
    pi_func x1 i1 = pi_func x2 i2 -> x1 = x2
}.Record partial_injection : Type := {
  pi_dom: Ensemble X;
  pi_func: forall x:X, In pi_dom x -> Y;
  pi_inj: forall (x1 x2:X) (i1:In pi_dom x1) (i2:In pi_dom x2),
    pi_func x1 i1 = pi_func x2 i2 -> x1 = x2
}.Record partial_injection : Type := {
  pi_dom: Ensemble X;
  pi_func: forall x:X, In pi_dom x -> Y;
  pi_inj: forall (x1 x2:X) (i1:In pi_dom x1) (i2:In pi_dom x2),
    pi_func x1 i1 = pi_func x2 i2 -> x1 = x2
}.Record partial_injection_ord (pi1 pi2:partial_injection) : Prop := {
  pi_dom_inc: Included (pi_dom pi1) (pi_dom pi2);
  pi_func_ext: forall (x:X) (i1:In (pi_dom pi1) x)
   (i2:In (pi_dom pi2) x),
    pi_func pi1 x i1 = pi_func pi2 x i2
}.Record partial_injection_ord (pi1 pi2:partial_injection) : Prop := {
  pi_dom_inc: Included (pi_dom pi1) (pi_dom pi2);
  pi_func_ext: forall (x:X) (i1:In (pi_dom pi1) x)
   (i2:In (pi_dom pi2) x),
    pi_func pi1 x i1 = pi_func pi2 x i2
}.Record partial_injection_ord (pi1 pi2:partial_injection) : Prop := {
  pi_dom_inc: Included (pi_dom pi1) (pi_dom pi2);
  pi_func_ext: forall (x:X) (i1:In (pi_dom pi1) x)
   (i2:In (pi_dom pi2) x),
    pi_func pi1 x i1 = pi_func pi2 x i2
}.

Lemma partial_injection_chain_ub: forall S:Ensemble partial_injection, chain partial_injection_ord S -> exists x:partial_injection, forall y:partial_injection, In S y -> partial_injection_ord y x.