Proof naturals_main_th3.
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(**************   FUNCTIONS         *******************)
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(********** in order to bound Domain and Range we first ********)
(********** define  Total   ***********)
Definition UnionPlus (F : Ens) := union F (Union F).
Definition Total (R : Ens) := UnionPlus (UnionPlus (UnionPlus R)).

(* Specifications: *)
Theorem UnionPlus_th1 : forall (X F : Ens) (i : IN X F), IN X (UnionPlus F).
Proof fun (X F : Ens) (i : IN X F) => union_th1 F (Union F) X i.
Theorem UnionPlus_th2 :
forall (X U F : Ens) (i : IN X U) (j : IN U F), IN X (UnionPlus F).