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Proof step 11 from proof lenequal (previous step)

# ax0 = # ay0 + # az, unique_al(ay0 + (a × d)' + az), unique_al(ax0), unique_al(ay0 + az) ∧ # ax0 = #(ay0 + az) ∧ (∀ a. a ∈ ax0 → a ∈ ay0 + az) ∧ b ∈ ay0 + az → b ∈ ax0, ∀ a0. a0 ∈ ax0 → a0 ∈ az ∨ a0 ∈ ay0 ⊦ b = a, a ∈ ax0, b ∈ ax0, a ∈ ay0, a ∈ az, ¬ b ∈ az ∧ ¬ b ∈ ay0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 12:
# ax0 = # ay0 + # az, unique_al(ay0 + (a × d)' + az), unique_al(ax0), ∀ a0. a0 ∈ ax0 → a0 ∈ az ∨ a0 ∈ ay0 ⊦ b = a, a ∈ ax0, b ∈ ax0, a ∈ ay0, a ∈ az, ¬ b ∈ az ∧ ¬ b ∈ ay0, (a0 ∈ ax0 → a0 ∈ ay0 ∨ a0 ∈ az) ∧ (b ∈ ay0 ∨ b ∈ az)

Rule argument

Additional information

Used simplifier rules: app: ⊦ unique_al(ax + p ' + ay) → unique_al(ax + ay);%inapp: ⊦ a ∈ ax + ay ↔ ¬ ¬(a ∈ ax ∨ a ∈ ay);% ⊦ #(ax + ax0) = # ax + # ax0;% ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1;% ⊦ m + n = n + m;