proposition: Lemma mcoeff_mpoly (E : 'X_{1..n} -> R) m k : mdeg m < k -> (\sum_(m : 'X_{1..n < k}) (E m *: 'X_[m]))@_m = E m.
proof: Proof. move=> lt_mk; rewrite raddf_sum (bigD1 (Sub m lt_mk)) //=. rewrite big1 ?addr0; last first. case=> i /= lt_ik; rewrite eqE /= => ne_im. by rewrite mcoeffZ mcoeffX (negbTE ne_im) mulr0. by rewrite mcoeffZ mcoeffX eqxx mulr1. Qed.
imports: HB(structures) mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat seq path) mathcomp(choice fintype tuple finfun bigop finset binomial) mathcomp(order fingroup perm ssralg zmodp poly ssrint) mathcomp(matrix vector) mathcomp(bigenough) ssrcomplements freeg