proposition: Lemma preimage_add_ker (f : {linear T -> U}) (A: set U) : [set a + b | a in f @^-1` A & b in f @^-1` [set 0]] = f @^-1` A.
proof: Proof. rewrite eqEsubset; split. - move=> x [a /= aA] [b /= bker] xe; subst x. by rewrite linearD bker addr0. - move=> x /= fx; exists x=>//. by exists 0; [ rewrite linear0 | rewrite addr0]. Qed.
imports: HB(structures) mathcomp(all_ssreflect ssralg fingroup perm matrix) mathcomp(mathcomp_extra boolp classical_sets Rstruct) mathcomp(ssrnum ereal) mathcomp(lra Rstruct reals) Reals ssrR Reals_ext Ranalysis_ext ssr_ext ssralg_ext logb realType_ext fdist