Use the definition of a linear space to show that
1. x + y = x + z ^ y = z
2. ax = ay and a ≠ 0 ⇔ x = y
3. ax = βx and x ≠ 0 ⇔ a = fi
4. (α - β)x = αx - βx
5. α(x - y) = αx - αy
6. α0 = 0
for all x, y, z ∈ X and α, β ∈ R.