import Mathlib.Data.Set.Function
variable {α β : Type _}
variable (f : α → β)
variable (s t : Set α)
open Set

example : f '' (s ∪ t) = f '' s ∪ f '' t :=
by sorry

theory Imagen_de_la_union
imports Main
begin

lemma "f ` (s ∪ t) = f ` s ∪ f ` t"
sorry

end

import Mathlib.Data.Set.Basic
import Mathlib.Tactic
open Set
variable {α : Type}
variable (s : Set α)
variable (A : ℕ → Set α)

example : s ∪ (⋂ i, A i) = ⋂ i, (A i ∪ s) :=
by sorry

theory Union_con_interseccion_general
imports Main
begin

lemma "s ∪ (⋂ i ∈ I. A i) = (⋂ i ∈ I. A i ∪ s)"
sorry

end

import Mathlib.Algebra.Ring.Parity

def Naturales : Set ℕ := {n | True}
def Pares     : Set ℕ := {n | Even n}
def Impares   : Set ℕ := {n | ¬Even n}

example : Pares ∪ Impares = Naturales :=
by sorry

theory Union_de_pares_e_impares
imports Main
begin

definition naturales :: "nat set" where
  "naturales = {n ∈ ℕ . True}"

definition pares :: "nat set" where
  "pares = {n ∈ ℕ . even n}"

definition impares :: "nat set" where
  "impares = {n ∈ ℕ . ¬ even n}"

lemma "pares ∪ impares = naturales"
oops

end

import Mathlib.Data.Set.Basic
import Mathlib.Tactic

open Set

variable {α : Type}
variable (s t : Set α)

example : (s \\ t) ∪ (t \\ s) = (s ∪ t) \\ (s ∩ t) :=
by sorry

theory Diferencia_de_union_e_interseccion
imports Main
begin

lemma "(s - t) ∪ (t - s) = (s ∪ t) - (s ∩ t)"
oops

end

import Mathlib.Data.Set.Basic
open Set

variable {α : Type}
variable (s t : Set α)

example : (s \\setminus t) ∪ t = s ∪ t :=
by sorry

theory Union_con_su_diferencia
imports Main
begin

lemma "(s - t) ∪ t = s ∪ t"
oops

end

import Mathlib.Data.Set.Basic
open Set
variable {α : Type}
variable (s t : Set α)

example : s ∪ (s ∩ t) = s :=
by sorry

theory Union_con_su_interseccion
imports Main
begin

lemma "s ∪ (s ∩ t) = s"
oops

end

import Mathlib.Data.Set.Basic
open Set
variable {α : Type}
variable (s t u : Set α)

example : s \ (t ∪ u) ⊆ (s \ t) \ u :=
by sorry

theory Diferencia_de_diferencia_de_conjuntos_2
imports Main
begin

lemma "(s - t) - u ⊆ s - (t ∪ u)"
oops

end

import Mathlib.Data.Set.Basic
import Mathlib.Tactic

open Set

variable {α : Type}
variable (s t u : Set α)

example : (s ∩ t) ∪ (s ∩ u) ⊆ s ∩ (t ∪ u):=
by sorry

theory Propiedad_semidistributiva_de_la_interseccion_sobre_la_union_2
imports Main
begin

lemma "(s ∩ t) ∪ (s ∩ u) ⊆ s ∩ (t ∪ u)"
oops

end

import Mathlib.Data.Set.Basic
import Mathlib.Tactic

open Set
variable {α : Type}
variable (s t u : Set α)

example : (s \ t) \ u ⊆ s \ (t ∪ u) :=
by sorry

theory Diferencia_de_diferencia_de_conjuntos
imports Main
begin

lemma "(s - t) - u ⊆ s - (t ∪ u)"
oops

import Mathlib.Data.Set.Basic
import Mathlib.Tactic
open Set
variable {α : Type}
variable (s t u : Set α)

example :
  s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u) :=
by sorry

theory Propiedad_semidistributiva_de_la_interseccion_sobre_la_union
imports Main
begin

lemma "s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u)"
oops

end