have : (∃ x, relate M β w x ∧ ¬evaluate M x ψ) := ⟨v, hyp⟩
ψ : Formula
W : Type
M : KripkeModel W
w : W
α β : Program
IHα : (∃ x, relate M α w x ∧ ¬evaluate M x ψ) ↔ ∃ f, (∃ a b, (a, b) ∈ H α ∧ Con (a ∪ [~⌈⌈b⌉⌉ψ]) = f) ∧ evaluate M w f
IHβ : (∃ x, relate M β w x ∧ ¬evaluate M x ψ) ↔ ∃ f, (∃ a b, (a, b) ∈ H β ∧ Con (a ∪ [~⌈⌈b⌉⌉ψ]) = f) ∧ evaluate M w f
v : W
hyp : relate M β w v ∧ ¬evaluate M v ψ
⊢ ∃ f, (∃ a b, ((a, b) ∈ H α ∨ (a, b) ∈ H β) ∧ Con (a ∪ [~⌈⌈b⌉⌉ψ]) = f) ∧ evaluate M w f
ψ : Formula
W : Type
M : KripkeModel W
w : W
α β : Program
IHα : (∃ x, relate M α w x ∧ ¬evaluate M x ψ) ↔ ∃ f, (∃ a b, (a, b) ∈ H α ∧ Con (a ∪ [~⌈⌈b⌉⌉ψ]) = f) ∧ evaluate M w f
IHβ : (∃ x, relate M β w x ∧ ¬evaluate M x ψ) ↔ ∃ f, (∃ a b, (a, b) ∈ H β ∧ Con (a ∪ [~⌈⌈b⌉⌉ψ]) = f) ∧ evaluate M w f
v : W
hyp : relate M β w v ∧ ¬evaluate M v ψ
this : ∃ x, relate M β w x ∧ ¬evaluate M x ψ
⊢ ∃ f, (∃ a b, ((a, b) ∈ H α ∨ (a, b) ∈ H β) ∧ Con (a ∪ [~⌈⌈b⌉⌉ψ]) = f) ∧ evaluate M w f