Set: pp::colors
  Set: pp::unicode
  Assumed: N
  Assumed: a
  Assumed: b
  Assumed: c
  Assumed: P
  Assumed: H3
  Proved: T1
  Proved: T2
  Proved: T3
  Proved: T4
Theorem T1 : ∃ x y z : N, P x y z :=
    ExistsIntro::explicit
        N
        (λ x : N, ∃ y z : N, P x y z)
        a
        (ExistsIntro::explicit
             N
             (λ x : N, if ((λ x::1 : N, if (P a x x::1) ⊥ ⊤) == (λ x : N, ⊤)) ⊥ ⊤)
             b
             (ExistsIntro::explicit N (λ z : N, P a b z) c H3))
Theorem T2 : ∃ x y z : N, P x y z := ExistsIntro a (ExistsIntro b (ExistsIntro c H3))
Theorem T3 : ∃ x y z : N, P x y z := ExistsIntro a (ExistsIntro b (ExistsIntro c H3))
Theorem T4 (H : P a a b) : ∃ x y z : N, P x y z := ExistsIntro a (ExistsIntro a (ExistsIntro b H))