Proof

Division (by d) with remainder shows that every polynomial f can be written in a unique way as the sum of a multiple of d and a polynomial of degree less than n (the remainder). This establishes the first claim.

The map f -> f mod d is linear since division with remainder applied to the polynomials f and g yields equalities

f = q₁d + r₁ and g = q₂d + r₂

from which we infer that for all elements a and b in R we have

af + bg = (aq₁ + bq₂)d + (ar₁ + br₂)

so that af + bg is mapped to ar₁ + br₂.

The kernel of the map consists of course precisely of all the multiples of d, and the image of the map is precisely R[X]_<n, since every such polynomial occurs as remainder upon division by d of that polynomial itself.