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'.

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.