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Proof step 11 from proof range-put (previous step)

n < # zbrar, n < nused, nused ≤ # zbrar, n ≠ 0 → zbrar[n - 1].key < key ⊦ zbrar[n].key < key, (key0 ∈ key-range(zbrar, n, nused, ri[key, adr]) ↔ key0 = key ∨ key0 ∈ key-range(zbrar, n, nused, ri)) ∧ (key0 ∈ key-range(zbrar, n, nused, ri[key, adr]) → ri[key, adr][key0] = key-range(zbrar, n, nused, ri)[key, adr][key0]), n ≠ 0 ∧ n + 1 < nused

User applied rule cut formula

And obtained the following new goals

Proof step 12:
n < nused, n < # zbrar, nused ≤ # zbrar, n ≠ 0 → zbrar[n - 1].key < key ⊦ n = 0, zbrar[n].key < key, n ≠ 0 ∧ n + 1 < nused, (key0 ∈ key-range(zbrar, n, nused, ri[key, adr]) ↔ key0 = key ∨ key0 ∈ key-range(zbrar, n, nused, ri)) ∧ (key0 ∈ key-range(zbrar, n, nused, ri[key, adr]) → ri[key, adr][key0] = key-range(zbrar, n, nused, ri)[key, adr][key0])
Proof step 17:
n = 0, n < nused, n < # zbrar, nused ≤ # zbrar, n ≠ 0 → zbrar[n - 1].key < key ⊦ zbrar[n].key < key, n ≠ 0 ∧ n + 1 < nused, (key0 ∈ key-range(zbrar, n, nused, ri[key, adr]) ↔ key0 = key ∨ key0 ∈ key-range(zbrar, n, nused, ri)) ∧ (key0 ∈ key-range(zbrar, n, nused, ri[key, adr]) → ri[key, adr][key0] = key-range(zbrar, n, nused, ri)[key, adr][key0])

Rule argument

fma: n ≠ 0

Additional information

Used formulas: n ≠ 0