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Set Universe Polymorphism.
Notation paths := identity .
Definition UU := Set.
Definition dirprod ( X Y : UU ) := sigT ( fun x : X => Y ) .
Definition dirprodpair { X Y : UU } := existT ( fun x : X => Y ) .
Definition hProp := sigT (fun X : Type => admit).
Definition hProptoType := @projT1 _ _ : hProp -> Type .
Coercion hProptoType: hProp >-> Sortclass.
Definition UU' := Type.
Definition hSet:= sigT (fun X : UU' => admit) .
Definition pr1hSet:= @projT1 UU (fun X : UU' => admit) : hSet -> Type.
Coercion pr1hSet: hSet >-> Sortclass.
Definition hrel ( X : UU ) := X -> X -> hProp.

Axiom hreldirprod : forall { X Y : UU } ( RX : hrel X ) ( RY : hrel Y ), hrel ( dirprod X Y ) .
Axiom iseqclass : forall { X : UU } ( R : hrel X ) ( A : hsubtypes X ), Type.