Proof for mathd_algebra_513

[aleph-prover-test]

Proven lemmas: 1/1

The goal is to prove that if real numbers a and b satisfy the linear system 3a + 2b = 5 and a + b = 2, then necessarily a = 1 and b = 1 (i.e., the unique solution is (1,1)). The proof is decomposed into two sub-goals: first derive a = 1 from the two equations, then derive 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 to obtain a = 1 and b = 1 directly from h₀ and h₁. With those in hand, the final step is just packaging the two equalities into a pair ⟨ha, hb⟩, which is also completed. Nothing remains unfinished; the theorem is fully proved. An interesting aspect is that the proof can be done either automatically via linarith (as here) or manually by eliminating one variable (e.g., substitute b = 2 − a into 3a + 2b = 5) to solve the system.