Proof for mathd_algebra_513

[aleph-prover-test]

Proven lemmas: 1/1

The goal is to prove that the real numbers a and b must both equal 1, assuming the linear equations 3a + 2b = 5 and a + b = 2. In other words, this is just solving a 2×2 system over ℝ and showing the unique solution is (1, 1).

The proof is decomposed into two subgoals because the conclusion is a conjunction: first prove a = 1, then prove b = 1. After splitting the goal, each part can be handled directly from the two hypotheses using linear arithmetic.

Progress is complete: 2 out of 2 sub-problems have been solved. The first part comes from combining the equations so that subtracting 2(a + b = 2) from 3a + 2b = 5 gives a = 1. Then b = 1 follows immediately from a + b = 2 together with a = 1, or again directly by linear arithmetic.

Nothing remains to be proved. An interesting feature is that Lean can dispatch both steps with linarith, since the whole argument is purely linear; alternatively, one could solve by substitution or note that the coefficient matrix has nonzero determinant, so the solution is unique.