 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
  M1 is locally_euclidean & M1,M2 are_homeomorphic implies
    M2 is locally_euclidean
proof
  assume that
A1: M1 is locally_euclidean
  and
A2: M1,M2 are_homeomorphic;
  let p be Point of M2;
  consider h be Function of M2,M1 such that
A3: h is being_homeomorphism by A2,T_0TOPSP:def 1;
  reconsider hp=h.p as Point of M1;
  consider U be a_neighborhood of hp,n such that
A4:M1|U,Tdisk(0.TOP-REAL n,1) are_homeomorphic by A1;
A5: rng h=[#]M1 by A3,TOPS_2:def 5;
  then
A6:h.:(h"U)=U by FUNCT_1:77;
  then reconsider hhU=h| (h"U) as Function of M2| (h"U),M1|U
    by JORDAN24:12;
A7: h"(Int U) c= h"U by TOPS_1:16,RELAT_1:143;
  hhU is being_homeomorphism by A3,A6,METRIZTS:2;
  then
A8: M2| (h"U), M1| U are_homeomorphic by T_0TOPSP:def 1;
  h"(Int U) is open by A5,A3,TOPS_2:43;
  then
A9:h"(Int U) c= Int (h"U) by A7,TOPS_1:24;
A10: dom h = [#]M2 by A3,TOPS_2:def 5;
  hp in Int U by CONNSP_2:def 1;
  then p in h"(Int U) by FUNCT_1:def 7,A10;
  then h"U is a_neighborhood of p by A9,CONNSP_2:def 1;
  hence thesis by A8,A4,BORSUK_3:3;
end;
