Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι → ο be given.
Assume H2:
and ((λ x4 : ι → ο . ∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . x5 x6 ⟶ x5 ((λ x7 : ι → ο . λ x8 . and (x7 x8) (x8 = prim0 (λ x9 . x7 x9) ⟶ ∀ x9 : ο . x9)) x6)) ⟶ (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . x6 x7 ⟶ x5 x7) ⟶ x5 (Descr_Vo1 x6)) ⟶ x5 x4) x3) (∀ x4 . x4 = x0 ⟶ x3 x4).
Claim L3: ∀ x4 . x4 = x1 ⟶ x3 x4
Let x4 of type ι be given.
Assume H3: x4 = x1.
Apply H3 with
λ x5 x6 . x3 x6.
Apply H0 with
x3.
The subproof is completed by applying H2.
Claim L4:
and ((λ x4 : ι → ο . ∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . x5 x6 ⟶ x5 ((λ x7 : ι → ο . λ x8 . and (x7 x8) (x8 = prim0 (λ x9 . x7 x9) ⟶ ∀ x9 : ο . x9)) x6)) ⟶ (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . x6 x7 ⟶ x5 x7) ⟶ x5 (Descr_Vo1 x6)) ⟶ x5 x4) x3) (∀ x4 . x4 = x1 ⟶ x3 x4)Apply andI with
(λ x4 : ι → ο . ∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . x5 x6 ⟶ x5 ((λ x7 : ι → ο . λ x8 . and (x7 x8) (x8 = prim0 (λ x9 . x7 x9) ⟶ ∀ x9 : ο . x9)) x6)) ⟶ (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . x6 x7 ⟶ x5 x7) ⟶ x5 (Descr_Vo1 x6)) ⟶ x5 x4) x3,
∀ x4 . x4 = x1 ⟶ x3 x4 leaving 2 subgoals.
Apply andEL with
(λ x4 : ι → ο . ∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . x5 x6 ⟶ x5 ((λ x7 : ι → ο . λ x8 . and (x7 x8) (x8 = prim0 (λ x9 . x7 x9) ⟶ ∀ x9 : ο . x9)) x6)) ⟶ (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . x6 x7 ⟶ x5 x7) ⟶ x5 (Descr_Vo1 x6)) ⟶ x5 x4) x3,
∀ x4 . x4 = x0 ⟶ x3 x4.
The subproof is completed by applying H2.
The subproof is completed by applying L3.
Apply H1 with
x3.
The subproof is completed by applying L4.