Section Set_Sums.
Variable U : Type.
Inductive Set_Sum (D : Ensemble (Ensemble U)) : U -> Prop :=
Set_Sum_intro :
forall B : Ensemble U,
In (Ensemble U) D B -> forall x : U, In U B x -> Set_Sum D x.
intros.
red in |- *; intros x x_in_Set_Sum_D.
red in x_in_Set_Sum_D.
elim x_in_Set_Sum_D.
assumption.
Qed.

Lemma Set_Sum_is_minoring :
forall (D : Ensemble (Ensemble U)) (A : Ensemble U),
In (Ensemble U) D A -> Included U A (Set_Sum D).