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 Th13:
  M is without_boundary n-locally_euclidean iff
  for p being Point of M holds
  ex U being a_neighborhood of p st U,[#]TOP-REAL n 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 Lm1,AA;
    take U;
    A3: (TOP-REAL n) | ([#]TOP-REAL n) = the TopStruct of TOP-REAL n
    by TSEP_1:93;
    reconsider B1 = (TOP-REAL n) | B as non empty TopSpace;
    M|U,B1 are_homeomorphic by A2,METRIZTS:def 1; then
    reconsider U1 = M|U as non empty TopSpace by YELLOW14:18;
    A4: U1,B1 are_homeomorphic by A2,METRIZTS:def 1;    
     B1, the TopStruct of TOP-REAL n are_homeomorphic
      by Th10,A3,METRIZTS:def 1;    
    hence U,[#]TOP-REAL n are_homeomorphic
     by A3,METRIZTS:def 1,A4,BORSUK_3:3;
  end;
  assume A6: for p being Point of M holds
  ex U being a_neighborhood of p st U,[#]TOP-REAL n are_homeomorphic;
  now
    let p be Point of M;
    consider U be a_neighborhood of p
    such that A7: U,[#]TOP-REAL n are_homeomorphic by A6;
    set S = the non empty ball Subset of TOP-REAL n;
    reconsider S as open Subset of TOP-REAL n;
    take U, S;
    A9: (TOP-REAL n) | ([#]TOP-REAL n) = the TopStruct of TOP-REAL n
    by TSEP_1:93;
    A10: M|U,the TopStruct of TOP-REAL n are_homeomorphic
    by A7,A9,METRIZTS:def 1; then
    reconsider U1 = M|U as non empty TopSpace by YELLOW14:18;
    reconsider S1 = (TOP-REAL n) | S as non empty TopSpace;
    A11: the TopStruct of TOP-REAL n, S1 are_homeomorphic
    by Th10,A9,METRIZTS:def 1;
    U1, S1 are_homeomorphic by A10,A11,BORSUK_3:3;
    hence U,S are_homeomorphic by METRIZTS:def 1;
  end;
  hence M is without_boundary n-locally_euclidean by Def4;
end;
