Proof for mathd_algebra_513

[aleph-prover-local]

Proven lemmas: 1/1

The goal is to prove that for real numbers a, b ∈ ℝ satisfying the linear equations 3·a + 2·b = 5 and a + b = 2, the unique solution is a = 1 and b = 1 (i.e., prove a = 1 ∧ b = 1). The proof is decomposed into two subgoals: first derive a = 1 from the two given equations, then derive b = 1 using the second equation together with the value of a. In Lean, this is done by defining ha : a = 1 and proving it with linarith from h₀ and h₁, then defining hb : b = 1 and proving it with linarith from h₁ and ha. Finally, the two results are combined into the required conjunction ⟨ha, hb⟩. At this point, all sub-problems are solved (2 out of 2), so the theorem is fully proved. An interesting aspect is that the proof uses automated linear arithmetic (linarith) rather than manual substitution, though substitution would also work as an alternative strategy.