proposition: Lemma closed_ball_subset (R : realFieldType) (M : normedModType R) (x : M) (r0 r1 : R) : 0 < r0 -> r0 < r1 -> closed_ball x r0 `<=` ball x r1.
proof: Proof. move=> r00 r01; rewrite (_ : r0 = (PosNum r00)%:num) // closed_ballE //. by move=> m xm; rewrite -ball_normE /ball_ /= (le_lt_trans _ r01). 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