 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
  for M be compact locally_euclidean non empty TopSpace st
     M is connected
  ex n st M is n-locally_euclidean
proof
  let M be compact locally_euclidean non empty TopSpace;
A1: for A be Subset of M st A is connected & [#]M c= A holds A=[#]M;
  assume M is connected;
  then [#]M is a_component by A1,CONNSP_1:27,CONNSP_1:def 5;
  then consider n such that
A2: M | [#]M is n-locally_euclidean non empty TopSpace by Th14;
  M | [#]M,M are_homeomorphic by Th1;
  then M is n-locally_euclidean by Th11,A2;
  hence thesis;
end;
