Real functions of two real variables

Uses Real numbers, Functions of several arguments

The sum, difference, product, and quotient of two functions is another function:
 ((ℝ, ℝ) → ℝ) + ((ℝ, ℝ) → ℝ) : (ℝ, ℝ) → ℝ
 ((ℝ, ℝ) → ℝ) - ((ℝ, ℝ) → ℝ) : (ℝ, ℝ) → ℝ
 ((ℝ, ℝ) → ℝ) × ((ℝ, ℝ) → ℝ) : (ℝ, ℝ) → ℝ
 ((ℝ, ℝ) → ℝ) ÷ ((ℝ, ℝ) → ℝ) : (ℝ, ℝ) → ℝ

Addition and multiplication are commutative and associative:
 f:(ℝ, ℝ) → ℝ, g:(ℝ, ℝ) → ℝ, f + g = g + f
 f:(ℝ, ℝ) → ℝ, g:(ℝ, ℝ) → ℝ, f × g = g × f
 f:(ℝ, ℝ) → ℝ, g:(ℝ, ℝ) → ℝ, h:(ℝ, ℝ) → ℝ, (f + g) + h = f + (g + h)
 f:(ℝ, ℝ) → ℝ, g:(ℝ, ℝ) → ℝ, h:(ℝ, ℝ) → ℝ, (f × g) × h = f × (g × h)

Addition is distributive over multiplication:
 f:(ℝ, ℝ) → ℝ, g:(ℝ, ℝ) → ℝ, h:(ℝ, ℝ) → ℝ, f × (g + h) = (f × g) + (f × h)

The sum, difference, product, and quotient of a function and a number is a function:
 ((ℝ, ℝ) → ℝ) + ℝ : (ℝ, ℝ) → ℝ
 ℝ + ((ℝ, ℝ) → ℝ) : (ℝ, ℝ) → ℝ
 ((ℝ, ℝ) → ℝ) - ℝ : (ℝ, ℝ) → ℝ
 ℝ - ((ℝ, ℝ) → ℝ) : (ℝ, ℝ) → ℝ
 ((ℝ, ℝ) → ℝ) × ℝ : (ℝ, ℝ) → ℝ
 ℝ × ((ℝ, ℝ) → ℝ) : (ℝ, ℝ) → ℝ
 ((ℝ, ℝ) → ℝ) ÷ ℝ : (ℝ, ℝ) → ℝ
 ℝ ÷ ((ℝ, ℝ) → ℝ) : (ℝ, ℝ) → ℝ

Addition and multiplication are commutative and associative:
 f:(ℝ, ℝ) → ℝ, s:ℝ, f + s = s + f
 f:(ℝ, ℝ) → ℝ, s:ℝ, f × s = s × f

Addition is distributive over multiplication:
 f:(ℝ, ℝ) → ℝ, g:(ℝ, ℝ) → ℝ, s:ℝ, s × (f + g) = (s × f) + (s × g)