Definition 76 (Landau 60). x + y = [Ξ₁ + Θ₁, Ξ₂ + Θ₂]. It is the sum of x and y: the operation of doing x and then y as addends.
Addition
Definitions 76–78 · Theorems 341–351
Throughout, x = [Ξ₁, Ξ₂], y = [Θ₁, Θ₂], z = [Λ₁, Λ₂].
Theorem 341 (Landau 209). x + y = y + x. (Commutative Law of Addition)
[Ξ₁ + Θ₁, Ξ₂ + Θ₂] = [Θ₁ + Ξ₁, Θ₂ + Ξ₂], by Theorem 309.
Theorem 342 (Landau 210). x + o = x.
[Ξ₁ + 0, Ξ₂ + 0] = [Ξ₁, Ξ₂].
Theorem 343 (Landau 211). (x + y) + z = x + (y + z). (Associative Law of Addition)
[(Ξ₁ + Θ₁) + Λ₁, (Ξ₂ + Θ₂) + Λ₂] = [Ξ₁ + (Θ₁ + Λ₁), Ξ₂ + (Θ₂ + Λ₂)], by Theorem 320.
Theorem 344 (Landau 212). For given x and y, the equation y + u = x has exactly one solution, u = [Ξ₁ − Θ₁, Ξ₂ − Θ₂].
Proof: With u = [Φ₁, Φ₂], the equation says Θ₁ + Φ₁ = Ξ₁ and Θ₂ + Φ₂ = Ξ₂. Theorem 321 does the rest.
Definition 77 (Landau 61). The u of Theorem 344 is the difference x − y.
Theorem 345 (Landau 213). x − y = o if and only if x = y.
Ξ₁ − Θ₁ = 0 and Ξ₂ − Θ₂ = 0 if and only if Ξ₁ = Θ₁ and Ξ₂ = Θ₂, by Theorem 316.
Definition 78 (Landau 62). −x = o − x.
Theorem 346 (Landau 214). −x = [−Ξ₁, −Ξ₂].
[0 − Ξ₁, 0 − Ξ₂] = [−Ξ₁, −Ξ₂].
Theorem 347 (Landau 215). −(−x) = x. Theorem 311.
Theorem 348 (Landau 216). x + (−x) = o. Theorem 313.
Theorem 349 (Landau 217). −(x + y) = −x + (−y).
[−(Ξ₁ + Θ₁), −(Ξ₂ + Θ₂)] = [−Ξ₁ + (−Θ₁), −Ξ₂ + (−Θ₂)], by Theorem 314.
Theorem 350 (Landau 218). x − y = x + (−y).
[Ξ₁ − Θ₁, Ξ₂ − Θ₂] = [Ξ₁, Ξ₂] + [−Θ₁, −Θ₂], by Definition 70.
Theorem 351 (Landau 219). −(x − y) = y − x.
−(x + (−y)) = −x + (−(−y)) = −x + y = y + (−x) = y − x.