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Proof step 44 from proof diff-union (previous step)

∀ y0, z0. rmdup(x0 + y0) \ z0 = rmdup((x0 \ z0) + (y0 \ z0)) ⊦ a ' + (rmdup(x0 + y) \ z) = a + rmdup((x0 \ z) + (y \ z)), a ∈ z, a ∈ x0, a ∈ y, a ∈ (x0 \ z) + (y \ z)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 45:
∀ y0, z0. rmdup(x0 + y0) \ z0 = rmdup((x0 \ z0) + (y0 \ z0)) ⊦ rmdup(x0 + y) \ z = rmdup((x0 \ z) + (y \ z)), a ∈ y, a ∈ x0, a ∈ z

Rule argument

Additional information

Used simplifier rules: diff-in: ⊦ a ∈ x \ y ↔ a ∈ x ∧ ¬ a ∈ y;% ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list% ⊦ x + y = x + z ↔ y = z; (s)% ⊦ a + x = a ' + x; (s)