Set: pp::colors
  Set: pp::unicode
  Imported 'Int'
  Assumed: a
  Assumed: P
  Assumed: f
  Assumed: g
  Assumed: H1
  Assumed: H2
  Assumed: H3
  Proved: T1
  Proved: T2
  Proved: T3
  Proved: T4
  Proved: T5
  Proved: T6
  Proved: T7
  Proved: T8
theorem T1 : ∃ x y : ℤ, P (f y x) (f y x) := ExistsIntro (g a) (ExistsIntro a H1)
theorem T2 : ∃ x : ℤ, P (f x (g x)) (f x (g x)) := ExistsIntro a H1
theorem T3 : ∃ x : ℤ, P (f x x) (f x x) := ExistsIntro (g a) H2
theorem T4 : ∃ x : ℤ, P (f (g a) x) (f x x) := ExistsIntro (g a) H2
theorem T5 : ∃ x : ℤ, P x x := ExistsIntro (f (g a) (g a)) H2
theorem T6 : ∃ x y : ℤ, P x y := ExistsIntro (f (g a) (g a)) (ExistsIntro (g a) H3)
theorem T7 : ∃ x : ℤ, P (f x x) x := ExistsIntro (g a) H3
theorem T8 : ∃ x y : ℤ, P (f x x) y := ExistsIntro (g a) (ExistsIntro (g a) H3)