 reserve n,m for Nat,
         p,q for Point of TOP-REAL n, r for Real;
reserve M,M1,M2 for non empty TopSpace;

theorem
  M is n-locally_euclidean & M is m-locally_euclidean implies n = m
proof
  assume that
A1: M is n-locally_euclidean
  and
A2: M is m-locally_euclidean;
  set p = the Point of M;
  consider W be a_neighborhood of p such that
A3:M|W,Tdisk(0.TOP-REAL m,1) are_homeomorphic by A2;
  consider U be a_neighborhood of p such that
A4:M|U,Tdisk(0.TOP-REAL n,1) are_homeomorphic by A1;
A5: p in Int W by CONNSP_2:def 1;
  p in Int U by CONNSP_2:def 1;
  hence thesis by A5,XBOOLE_0:3,A4,A3,BROUWER3:14;
end;
