Variable V : Type.

(* Specifications: *)
Theorem Image_set_continuous :
forall (A:Ensemble U) (f:U -> V) (X:Ensemble V),
Finite V X ->
Included V X (Im U V A f) ->
exists n : _,
(exists Y : _, (cardinal U Y n /\ Included U Y A) /\ Im U V Y f = X).
Theorem Image_set_continuous' :
forall (A:Ensemble U) (f:U -> V) (X:Ensemble V),
Approximant V (Im U V A f) X ->
exists Y : _, Approximant U A Y /\ Im U V Y f = X.
Theorem Pigeonhole_bis :
forall (A:Ensemble U) (f:U -> V),
~ Finite U A -> Finite V (Im U V A f) -> ~ injective U V f.
Theorem Pigeonhole_ter :
forall (A:Ensemble U) (f:U -> V) (n:nat),
injective U V f -> Finite V (Im U V A f) -> Finite U A.