reserve n for Nat;
reserve p for Point of TOP-REAL n, r for Real;
reserve q for Point of TOP-REAL n;

theorem Th10:
  for B being non empty ball Subset of TOP-REAL n
  holds B, [#]TOP-REAL n are_homeomorphic
proof
  let B be non empty ball Subset of TOP-REAL n;
  consider p be Point of TOP-REAL n, r be Real such that
  A1: B = Ball(p,r) by Def1;
  reconsider B1 = Tball(p,r) as non empty TopSpace by A1;
  A2: TOP-REAL n, (TOP-REAL n) | ([#]TOP-REAL n) are_homeomorphic by MFOLD_0:1;
  A3: Tunit_ball n, TOP-REAL n are_homeomorphic by Th8;
  r is positive by A1; then
  Tball(p,r), Tball(0.(TOP-REAL n),1) are_homeomorphic by MFOLD_0:3; then
  B1, TOP-REAL n are_homeomorphic by A3,BORSUK_3:3;
::  Tball(p,r) , (TOP-REAL n) | ([#]TOP-REAL n) are_homeomorphic
::  by A2,BORSUK_3:3;
  hence B, [#]TOP-REAL n are_homeomorphic by A1,METRIZTS:def 1,A2,BORSUK_3:3;
end;
