Proof for mathd_algebra_513

[aleph-prover-test]

Proven lemmas: 1/1

The goal is to prove that for real numbers a, b ∈ ℝ satisfying the linear system 3a + 2b = 5 and a + b = 2, the unique solution is a = 1 and b = 1 (i.e., to prove the conjunction a = 1 ∧ b = 1). The proof is decomposed by splitting the conjunction into two subgoals: first show a = 1, then show b = 1. In Lean this is done with constructor, which creates exactly these two goals. Both subgoals are solved using linarith with the hypotheses h₀ and h₁, which automatically performs the required linear algebra reasoning. So far, all sub-problems are completed: 2 goals out of 2 have been solved, and the theorem is finished. Nothing remains to be proved. An interesting aspect is that the entire “solve the 2×2 system” step can be delegated to linarith, though an alternative manual strategy would be to eliminate one variable using a + b = 2, substitute into 3a + 2b = 5, and solve.