Proof for mathd_algebra_513

[aleph-prover-test]

Proven lemmas: 1/1

The current goal is to prove that if real numbers a, b ∈ ℝ satisfy the system 3a + 2b = 5 and a + b = 2, then necessarily a = 1 and b = 1. In other words, Lean is proving the unique solution of this 2×2 linear system.

The statement was naturally split into two sub-problems because the conclusion is a conjunction: first prove a = 1, then prove b = 1. Both of these sub-goals have already been solved.

So the progress is complete: 2 out of 2 sub-problems are finished, and the theorem is essentially done. There is no remaining mathematical work on this proof.

The proof strategy is straightforward linear elimination. One can either let linarith solve each variable directly from the two equations, or argue by hand: subtract 2(a + b = 2) from 3a + 2b = 5 to get a = 1, then substitute back into a + b = 2 to obtain b = 1. An interesting point is that the coefficient matrix has nonzero determinant, so the solution is unique, which matches the result (a, b) = (1, 1).