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Proof step 7 from proof firstoccurrence-key (previous step)

¬ a ∈ ax0, ∀ a. a ∈ ax0 → (∃ ax2, d, ax1. ax0 = ax2 + (a × d)' + ax1 ∧ ¬ a ∈ ax2) ⊦ ∃ ax, d0, ax1. (a0 × d)' + ax0 = ax + (a × d0)' + ax1 ∧ ¬ a ∈ ax, a ≠ a0 ∧ ¬ a ∈ ax0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
∀ a. a ∈ ax0 → (∃ ax2, d, ax1. ax0 = ax2 + (a × d)' + ax1 ∧ ¬ a ∈ ax2) ⊦ a0 ∈ ax0, ∃ ax, d0, ax1. (a0 × d)' + ax0 = ax + (a0 × d0)' + ax1 ∧ ¬ a0 ∈ ax

Proof step 7 from proof firstoccurrence-key (previous step)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information