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Proof step 11 from proof cont-monotone (previous step)

f(S0)(x), S0 ⊆ S, chain(λ n. (n = 0 ⊃ S0;S)) → (∀ x. f(λ x0. ∃ n. (n = 0 ⊃ S0;S)(x0))(x) ↔ (∃ n. f((n = 0 ⊃ S0;S))(x))) ⊦ f(S)(x)

User applied rule case distinction

And obtained the following new goals

Proof step 12:
f(S0)(x), S0 ⊆ S ⊦ chain(λ n. (n = 0 ⊃ S0;S)), f(S)(x)
Proof step 20:
∀ x. f(λ x0. ∃ n. (n = 0 ⊃ S0;S)(x0))(x) ↔ (∃ n. f((n = 0 ⊃ S0;S))(x)), f(S0)(x), S0 ⊆ S ⊦ f(S)(x)

Rule argument

pos: left 3

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