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Proof step 3 from proof put-in-01 (previous step)

m ≤ # x0, ∀ m, a, b. m < # x0 → (a ∈ x0[m, b] ↔ a = b ∨ a ∈ x0 -1l x0[m]) ⊦ a ∈ (a0 ' + x0)[m, b] ↔ a = b ∨ a ∈ (a0 ' + x0) -1l (a0 ' + x0)[m]

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 4:
(a0 ' + x0)[m] = a0, m ≤ # x0, ∀ m, a, b. m < # x0 → (a ∈ x0[m, b] ↔ a = b ∨ a ∈ x0 -1l x0[m]) ⊦ a ∈ (a0 ' + x0)[m, b] ↔ a = b ∨ a ∈ (a0 ' + x0) -1l (a0 ' + x0)[m]
Proof step 9:
m ≤ # x0, ∀ m, a, b. m < # x0 → (a ∈ x0[m, b] ↔ a = b ∨ a ∈ x0 -1l x0[m]) ⊦ (a0 ' + x0)[m] = a0, a ∈ (a0 ' + x0)[m, b] ↔ a = b ∨ a ∈ (a0 ' + x0) -1l (a0 ' + x0)[m]

Rule argument

fma: (a0 ' + x0)[m] = a0

Additional information

Used formulas: (a0 ' + x0)[m] = a0