proposition: Lemma scale_ballE k x r : 0 <= k -> k *` ball x r = ball x (k * r).
proof: Proof. move=> k0; have [r0|r0] := ltP 0 r. apply/seteqP; split => y. rewrite /scale_ball is_ball_ball//= cpoint_ball//. by rewrite (radius_ball_num _ (ltW _)). by rewrite /scale_ball is_ball_ball cpoint_ball// radius_ball_num// ltW. rewrite ((ball0 _ _).2 r0) scale_ball_set0; apply/esym/ball0. by rewrite mulr_ge0_le0. Qed.
imports: HB(structures) mathcomp(all_ssreflect ssralg ssrint ssrnum finmap matrix) mathcomp(rat interval zmodp vector fieldext falgebra) mathcomp(mathcomp_extra boolp classical_sets functions) mathcomp(cardinality set_interval Rstruct) ereal reals signed topology prodnormedzmodule