reserve n for Nat;
reserve p for Point of TOP-REAL n, r for Real;
reserve q for Point of TOP-REAL n;
reserve M for non empty TopSpace;

theorem
  M is without_boundary n-locally_euclidean iff
  for p being Point of M holds
  ex U being a_neighborhood of p, B being ball Subset of TOP-REAL n
    st U,B are_homeomorphic
proof
  hereby
    assume M is without_boundary n-locally_euclidean;
then AA:for p being Point of M holds
      ex U being a_neighborhood of p, S being open Subset of TOP-REAL n
    st U,S are_homeomorphic by Def4;
    let p be Point of M;
    consider U be a_neighborhood of p, B be non empty ball Subset of TOP-REAL n
    such that A2: U,B are_homeomorphic by AA,Lm1;
    reconsider B as ball Subset of TOP-REAL n;
    take U, B;
    thus U, B are_homeomorphic by A2;
  end;
  assume A3: for p being Point of M holds
  ex U being a_neighborhood of p, B being ball Subset of TOP-REAL n
  st U,B are_homeomorphic;
  now
    let p be Point of M;
    consider U be a_neighborhood of p, B be ball Subset of TOP-REAL n
    such that A4: U,B are_homeomorphic by A3;
    reconsider S = B as open Subset of TOP-REAL n;
    take U, S;
    thus U,S are_homeomorphic by A4;
  end;
  hence M is without_boundary n-locally_euclidean by Def4;
end;
