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Proof step 29 from proof cont-monotone (previous step)

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

User applied rule case distinction

And obtained the following new goals

Proof step 30:
∃ n. (n = 0 ⊃ S0;S)(x0), f(S0)(x), S0 ⊆ S, f(λ x0. ∃ n. (n = 0 ⊃ S0;S)(x0))(x) ⊦ S(x0), f(S)(x)
Proof step 36:
S(x0), f(S0)(x), S0 ⊆ S, f(λ x0. ∃ n. (n = 0 ⊃ S0;S)(x0))(x) ⊦ ∃ n. (n = 0 ⊃ S0;S)(x0), f(S)(x)

Rule argument

pos: right 1

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