 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 Th13:
  M is without_boundary n-locally_euclidean non empty TopSpace
    iff
  for p be Point of M ex U be a_neighborhood of p st
    M|U,Tball(0.TOP-REAL n,1) are_homeomorphic
proof
  set TR=TOP-REAL n;
  hereby
    assume
A1: M is without_boundary n-locally_euclidean non empty TopSpace;
    then reconsider MM=M as without_boundary n-locally_euclidean
      non empty TopSpace;
    let p be Point of M;
    consider U be a_neighborhood of p such that
A2: M|U,Tdisk(0.TR,1) are_homeomorphic by Def3,A1;
    set MU=M|U;
    consider h be Function of MU,Tdisk(0.TR,1) such that
A3: h is being_homeomorphism by A2,T_0TOPSP:def 1;
A4: [#]Tdisk(0.TR,1) = cl_Ball(0.TR,1) by PRE_TOPC:def 5;
    then reconsider B=Ball(0.TR,1) as Subset of Tdisk(0.TR,1)
      by TOPREAL9:16;
    reconsider B as open Subset of Tdisk(0.TR,1) by TSEP_1:9;
A5: Int U c= U by TOPS_1:16;
    set HIB = B /\ h.:(Int U);
    reconsider hib=HIB as Subset of TR by A4,XBOOLE_1:1;
A6: HIB c= Ball(0.TR,1) by XBOOLE_1:17;
A7: h"HIB c= h"(h.:Int U) by XBOOLE_1:17,RELAT_1:143;
A8: h"(h.:Int U) c= Int U by A3,FUNCT_1:82;
A9: cl_Ball(0.TR,1) = Ball(0.TR,1) \/ Sphere(0.TR,1) by TOPREAL9:18;
A10:[#]MU = U by PRE_TOPC:def 5;
    then reconsider IU=Int U as Subset of M|U by TOPS_1:16;
A11:p in Int U by CONNSP_2:def 1;
A12: dom h = [#]MU by A3,TOPS_2:def 5;
    then
A13: h.p in rng h by A11,A5,A10,FUNCT_1:def 3;
A14: h.p in Ball(0.TR,1)
    proof
      assume not h.p in Ball(0.TR,1);
      then h.p in Sphere(0.TR,1) by A4,A13,A9,XBOOLE_0:def 3;
      then p in Fr MM by A3,Th5;
      hence contradiction;
    end;
    then reconsider hP=h.p as Point of TR;
    IU is open by TSEP_1:9;
    then h.:(Int U) is open by A3,TOPGRP_1:25;
    then
A15: hib is open by TSEP_1:9,A6;
    h.p in h.:(Int U) by A11,A5,A10,A12,FUNCT_1:def 6;
    then h.p in hib by A14,XBOOLE_0:def 4;
    then consider r be positive Real such that
A16:   Ball(hP,r) c= hib by A15,TOPS_4:2;
    |.hP-hP.|=0 by TOPRNS_1:28;
    then A17:hP in Ball(hP,r);
    reconsider bb=Ball(hP,r) as non empty Subset of Tdisk(0.TR,1)
      by A16,XBOOLE_1:1;
    reconsider hb=h"bb as Subset of M by A10,XBOOLE_1:1;
    bb is open by TSEP_1:9;
    then
A18:h"bb is open by A3,TOPGRP_1:26;
A19:M|hb = M|U | (h"bb) by A10,PRE_TOPC:7;
    hb c= h"HIB by A16,RELAT_1:143;
    then hb c= Int U by A7,A8;
    then
A20:hb is open by A10,TSEP_1:9,A18,A5;
    p in hb by A17, A11,A5,A10,A12,FUNCT_1:def 7;
    then
A21: p in Int hb by A20,TOPS_1:23;
    rng h = [#]Tdisk(0.TR,1) by A3,TOPS_2:def 5;
    then
A22: h.: (h"bb) = bb by FUNCT_1:77;
A24: Tdisk(0.TR,1) | bb = TR|Ball(hP,r) by A4,PRE_TOPC:7;
    then reconsider hh=h| (h"bb) as Function of M|hb,Tball(hP,r)
      by A19,JORDAN24:12,A22;
    reconsider hb as a_neighborhood of p by A21,CONNSP_2:def 1;
    hh is being_homeomorphism by A3,A19,A22,A24,METRIZTS:2;
    then
A25:  M|hb,Tball(hP,r) are_homeomorphic by T_0TOPSP:def 1;
    take hb;
    Tball(hP,r),Tball(0.TR,1) are_homeomorphic by Th3;
    hence M|hb,Tball(0.TR,1) are_homeomorphic by A25,BORSUK_3:3;
  end;
  assume
A26:for p be Point of M ex U being a_neighborhood of p st
    M|U,Tball(0.TR,1) are_homeomorphic;
  M is n-locally_euclidean
  proof
    [#]Tdisk(0.TR,1) = cl_Ball(0.TR,1) by PRE_TOPC:def 5;
    then reconsider B=Ball(0.TR,1) as Subset of Tdisk(0.TR,1)
      by TOPREAL9:16;
    reconsider B as open Subset of Tdisk(0.TR,1) by TSEP_1:9;
    let p be Point of M;
    consider U be a_neighborhood of p such that
A27: M|U,Tball(0.TR,1) are_homeomorphic by A26;
A28:  n in NAT by ORDINAL1:def 12;
A30: [#]Tball(0.TR,1) = Ball(0.TR,1) by PRE_TOPC:def 5;
    then reconsider clB = cl_Ball(0.TR,1/2) as non empty Subset of
      Tball(0.TR,1) by JORDAN:21,A28;
    set MU=M|U;
    consider h be Function of MU,Tball(0.TR,1) such that
A31:  h is being_homeomorphism by A27,T_0TOPSP:def 1;
    set En=Euclid n;
A32: Int U c= U by TOPS_1:16;
A33: [#]MU = U by PRE_TOPC:def 5;
    then reconsider IU=Int U as Subset of M|U by TOPS_1:16;
    reconsider hIU = h.:IU as Subset of TR by A30,XBOOLE_1:1;
A34: dom h = [#]MU by A31,TOPS_2:def 5;
    then
A35: IU=h"hIU by A31,FUNCT_1:82,FUNCT_1:76;
A36:the TopStruct of TR = TopSpaceMetr En by EUCLID:def 8;
    then reconsider hIUE=hIU as Subset of TopSpaceMetr En;
A37:p in Int U by CONNSP_2:def 1;
    then
A38: h.p in hIU by A34,FUNCT_1:def 6;
    then reconsider hp=h.p as Point of En by EUCLID:67;
    reconsider Hp=h.p as Point of TR by A38;
    IU is open by TSEP_1:9;
    then h.:IU is open by A31,TOPGRP_1:25;
    then hIU is open by A30,TSEP_1:9;
    then hIU in the topology of TR by PRE_TOPC:def 2;
    then consider r be Real such that
A39: r > 0
    and
A40: Ball(hp,r) c= hIUE by A38,A36,PRE_TOPC:def 2,TOPMETR:15;
    set r2=r/2;
A41: Ball(Hp,r)=Ball(hp,r) by TOPREAL9:13;
    cl_Ball(Hp,r2) c= Ball(Hp,r) by A28,XREAL_1:216,A39,JORDAN:21;
    then
A42: cl_Ball(Hp,r2) c= hIU by A40,A41;
    then
    reconsider CL=cl_Ball(Hp,r2) as Subset of Tball(0.TR,1) by XBOOLE_1:1;
A43: cl_Ball(Hp,r2) c= [#]Tball(0.TR,1) by A42,XBOOLE_1:1;
    then cl_Ball(Hp,r2) c= rng h by A31,TOPS_2:def 5;
    then
A44:  h.:(h"CL) = CL by FUNCT_1:77;
    set r22=r2/2;
    r22 < r2 by A39,XREAL_1:216;
    then
A45: Ball(Hp,r22) c= Ball(Hp,r2) by A28,JORDAN:18;
    reconsider hCL=h"CL as Subset of M by A33,XBOOLE_1:1;
A46: (M|U) | (h"CL) = M| hCL by A33,PRE_TOPC:7;
A47: Ball(Hp,r2) c= cl_Ball(Hp,r2) by TOPREAL9:16;
    then
A48:Ball(Hp,r22) c= cl_Ball(Hp,r2) by A45;
    then reconsider BI = Ball(Hp,r22) as Subset of Tball(0.TR,1)
      by A43,XBOOLE_1:1;
    BI c= hIU by A42,A47,A45;
    then
A49: h"BI c= h"hIU by RELAT_1:143;
    reconsider hBI=h"BI as Subset of M by A33,XBOOLE_1:1;
    BI is open by TSEP_1:9;
    then h"BI is open by A31,TOPGRP_1:26;
    then
A50: hBI is open by A35,A49,TSEP_1:9;
    |.Hp - Hp.|=0 by TOPRNS_1:28;
    then h.p in BI by A39;
    then p in h"BI by A37,A32,A33,A34,FUNCT_1:def 7;
    then p in Int hCL by A48,RELAT_1:143,TOPS_1:22,A50;
    then reconsider hCL as a_neighborhood of p by CONNSP_2:def 1;
A51: Tball(0.TR,1) |CL = Tdisk(Hp,r2) by A30,PRE_TOPC:7;
    then reconsider hh=h| (h"CL) as Function of M|hCL ,Tdisk(Hp,r2)
      by A44,JORDAN24:12,A46;
    hh is being_homeomorphism by A31,A51,METRIZTS:2,A44,A46;
    then
A52:M|hCL,Tdisk(Hp,r2) are_homeomorphic by T_0TOPSP:def 1;
    Tdisk(Hp,r2),Tdisk(0.TR,1) are_homeomorphic by Lm1,A39;
    hence thesis by A52,BORSUK_3:3,A39;
  end;
  then reconsider M as n-locally_euclidean non empty TopSpace;
  M is without_boundary
  proof
    thus Int M c= the carrier of M;
    let x be object;
    assume x in the carrier of M;
    then reconsider p=x as Point of M;
    ex U being a_neighborhood of p st
      M|U,Tball(0.TOP-REAL n,1) are_homeomorphic by A26;
    hence thesis by Def4;
  end;
  hence thesis;
end;
