 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 Th11:
  M1 is n-locally_euclidean & M2 is locally_euclidean &
    M1,M2 are_homeomorphic implies M2 is n-locally_euclidean
proof
  assume that
A1: M1 is n-locally_euclidean
  and
A2: M2 is locally_euclidean
  and
A3: M1,M2 are_homeomorphic;
  consider h be Function of M1,M2 such that
A4: h is being_homeomorphism by A3,T_0TOPSP:def 1;
  let q be Point of M2;
  set hp=q;
  consider W be a_neighborhood of hp,m such that
A5:M2|W,Tdisk(0.TOP-REAL m,1) are_homeomorphic by A2;
A6: rng h=[#]M2 by A4,TOPS_2:def 5;
  then
A7: h"W is non empty by RELAT_1:139;
A8: dom h = [#]M1 by A4,TOPS_2:def 5;
  then consider p be object such that
A9: p in [#]M1
  and
A10: h.p=q by A6,FUNCT_1:def 3;
  reconsider p as Point of M1 by A9;
  consider U be a_neighborhood of p such that
A11:M1|U,Tdisk(0.TOP-REAL n,1) are_homeomorphic by A1;
  consider h2 be Function of M2|W,Tdisk(0.TOP-REAL m,1) such that
A12: h2 is being_homeomorphism by T_0TOPSP:def 1,A5;
A13:h.:(h"W)=W by A6,FUNCT_1:77;
  then reconsider hhW=h| (h"W) as Function of M1| (h"W), M2| W
    by JORDAN24:12;
  hhW is being_homeomorphism by A4, A13,METRIZTS:2;
  then h2*hhW is being_homeomorphism by A7,A12,TOPS_2:57;
  then
A14:M1| (h"W),Tdisk(0.TOP-REAL m,1) are_homeomorphic by T_0TOPSP:def 1;
A15: h"(Int W) c= h"W by TOPS_1:16,RELAT_1:143;
  h"(Int W) is open by A6, A4,TOPS_2:43;
  then
A16:h"(Int W) c= Int (h"W) by A15,TOPS_1:24;
  hp in Int W by CONNSP_2:def 1;
  then
A17:p in h"(Int W) by A10,FUNCT_1:def 7,A8;
  p in Int U by CONNSP_2:def 1;
  then n=m by A17,A16,XBOOLE_0:3,A14,A11,BROUWER3:14;
  hence thesis by A5;
end;
