reserve T1,T2,T3 for TopSpace,
  A1 for Subset of T1, A2 for Subset of T2, A3 for Subset of T3;
reserve n,k for Nat;
reserve M,N for non empty TopSpace;

theorem Th12:
  M is n-manifold & M,N are_homeomorphic implies N is n-manifold
proof
  assume
A1: M is n-manifold;
  assume
A2: M,N are_homeomorphic;
  then
A3: N is second-countable by A1,Th9;
A4: N is Hausdorff by A1,A2,Th10;
  N is n-locally_euclidean by A1,A2,MFOLD_0:10,11;
  hence N is n-manifold by A3,A4;
end;
