Section Evar_env''LULSU.
Variable L U : nat.
Definition Evar_env'LU : Ensemble var_env' := Evar_env' L U.
Definition Evar_env''LU : Ensemble var_env'' := Evar_env'' L U.
Definition Evar_env'LSU : Ensemble var_env' := Evar_env' L (S U).
Definition Evar_env''LSU : Ensemble var_env'' := Evar_env'' L (S U).
Definition f1 : var_env' -> var_env' := Evar_env'ntoSn U.
Definition f1' : var_env'' -> var_env'' := Evar_env''ntoSn U.
Definition imagef1 : Ensemble var_env' := Im _ _ Evar_env'LU f1.
Definition imagef1' : Ensemble var_env'' := Im _ _ Evar_env''LU f1'.
Definition f2 (x : var_env') : var_env' := x.
Definition f2' (x : var_env'') : var_env'' := x.
Definition imagef2 : Ensemble var_env' := Im _ _ Evar_env'LU f2.
Definition imagef2' : Ensemble var_env'' := Im _ _ Evar_env''LU f2'.
Definition imagef1orf2 : Ensemble var_env' := Union _ imagef1 imagef2.
Definition imagef1'orf2' : Ensemble var_env'' := Union _ imagef1' imagef2'.

(* Specifications: *)
Lemma var_env''M0 :
forall ve : var_env'', U - L = 0 -> In _ Evar_env''LU ve -> ve = M0 _.
Lemma same_set_same_cardinal :
forall (U : Type) (X Y : Ensemble U) (n m : nat),
cardinal U X n -> cardinal U Y m -> Same_set U X Y -> n = m.
Lemma same_set_finite :
forall (U : Type) (X Y : Ensemble U),
Finite U X -> Same_set U X Y -> Finite U Y.
Lemma singleton_add_empty :
forall (U : Type) (x : U),
Same_set _ (Singleton _ x) (Add _ (Empty_set _) x).
Lemma singleton_cardinal_one :
forall (U : Type) (x : U), cardinal _ (Singleton _ x) 1.
Lemma M0inEvar_env'' : In (Map unit) Evar_env''LU (M0 unit).
Lemma var_env''singleton :
U - L = 0 -> Same_set _ Evar_env''LU (Singleton _ (M0 _)).
Lemma var_env''cardinal_one : U - L = 0 -> cardinal _ Evar_env''LU 1.
Lemma imagef1lemma' :
forall ve : var_env'',
In _ Evar_env''LSU ve ->
in_dom _ (N_of_nat U) ve = true ->
exists ve' : var_env'', In _ Evar_env''LU ve' /\ f1' ve' = ve.
Lemma imagef2lemma' :
forall ve : var_env'',
In _ Evar_env''LSU ve ->
in_dom _ (N_of_nat U) ve = false ->
exists ve' : var_env'', In _ Evar_env''LU ve' /\ f2' ve' = ve.
Lemma imagef1'orf2'lemma :
forall ve : var_env'', In _ Evar_env''LSU ve -> In _ imagef1'orf2' ve.