Proof for mathd_algebra_513

[aleph-prover-test]

Proven lemmas: 1/1

The goal is to prove that if a, b ∈ ℝ satisfy the two linear equations 3·a + 2·b = 5 and a + b = 2, then necessarily a = 1 and b = 1 (i.e., the unique solution to this 2×2 system is (1,1)).
The proof is decomposed into two sub-goals: first derive ha : a = 1 from the hypotheses, then derive hb : b = 1 from the same hypotheses, and finally combine them into the conjunction a = 1 ∧ b = 1.
So far, both sub-goals have been solved: Lean uses the linear arithmetic tactic linarith with [h₀, h₁] to compute a = 1, and similarly to compute b = 1.
With ha and hb in hand, the final step is just to package them as ⟨ha, hb⟩, which is also completed.
Nothing remains: the theorem is fully proved.
An interesting aspect is that the proof avoids manual algebraic substitution; linarith automatically solves the linear system, though a manual alternative (solve for b from a + b = 2, substitute into 3a + 2b = 5) is also noted as a viable strategy.