 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 Th5:
  for M be locally_euclidean non empty TopSpace
    for p be Point of M holds
        p in Fr M
     iff
        ex U being a_neighborhood of p,
           n be Nat,
           h be Function of M|U,Tdisk(0.TOP-REAL n,1)
        st h is being_homeomorphism & h.p in Sphere(0.TOP-REAL n,1)
proof
  let M be locally_euclidean non empty TopSpace;
  let p be Point of M;
  thus p in Fr M implies ex U be a_neighborhood of p,n be Nat,
    h be Function of M| U,Tdisk(0.TOP-REAL n,1) st
    h is being_homeomorphism & h.p in Sphere(0.TOP-REAL n,1)
  proof
    assume
A1: p in Fr M;
    consider U be a_neighborhood of p, n such that
A2: M|U,Tdisk(0.TOP-REAL n,1) are_homeomorphic by Def2;
    set TR=TOP-REAL n;
    consider h be Function of M|U,Tdisk(0.TR,1) such that
A3: h is being_homeomorphism by A2,T_0TOPSP:def 1;
    assume for U be a_neighborhood of p,n be Nat, h be Function of M|U,
      Tdisk(0.TOP-REAL n,1) st h is being_homeomorphism holds
      not h.p in Sphere(0.TOP-REAL n,1);
    then
A4: not h.p in Sphere(0.TR,1) by A3;
A5: Ball(0.TR,1) in the topology of TR by PRE_TOPC:def 2;
A6: p in Int U by CONNSP_2:def 1;
A7: Int U in the topology of M by PRE_TOPC:def 2;
A8: [#](M|U) = U by PRE_TOPC:def 5;
    then reconsider IU=Int U as Subset of M|U by TOPS_1:16;
    IU/\U = IU by TOPS_1:16,XBOOLE_1:28;
    then IU in the topology of M|U by A7,A8,PRE_TOPC:def 4;
    then reconsider IU as non empty open Subset of M|U
      by CONNSP_2:def 1,PRE_TOPC:def 2;
A9: [#](TR|cl_Ball(0.TR,1)) = cl_Ball(0.TR,1) by PRE_TOPC:def 5;
    then reconsider hIU=h.:IU as Subset of TR by XBOOLE_1:1;
A10: h.:IU is open by A3,TOPGRP_1:25;
A11: dom h = [#](M|U) by A3,TOPS_2:def 5;
    then
A12: h"(h.:IU) = IU by FUNCT_1:94,A3;
A13: cl_Ball(0.TR,1) = Ball(0.TR,1) \/ Sphere(0.TR,1) by TOPREAL9:18;
    then reconsider B=Ball(0.TR,1) as Subset of TR|cl_Ball(0.TR,1)
      by A9,XBOOLE_1:7;
    Ball(0.TR,1) /\ cl_Ball(0.TR,1) = Ball(0.TR,1) by A13,XBOOLE_1:7,28;
    then B in the topology of TR|cl_Ball(0.TR,1) by A5,A9,PRE_TOPC:def 4;
    then reconsider B as non empty open Subset of TR|cl_Ball(0.TR,1)
      by PRE_TOPC:def 2;
    reconsider BhIU = B /\ (h.:IU) as Subset of TR by XBOOLE_1:1,A9;
    BhIU c= Ball(0.TR,1) by XBOOLE_1:17;
    then
A14: BhIU is open by A10,TSEP_1:9;
A15: Int U c= U by TOPS_1:16;
    then h.p in rng h by A6,A8,A11,FUNCT_1:def 3;
    then
A16:h.p in Ball(0.TR,1) by A4,A9,A13,XBOOLE_0:def 3;
    then reconsider hp=h.p as Point of TR;
    the TopStruct of TR=TopSpaceMetr Euclid n by EUCLID:def 8;
    then reconsider HP=hp as Point of Euclid n by TOPMETR:12;
    h.p in h.:IU by A11,A6,FUNCT_1:def 6;
    then h.p in BhIU by A16,XBOOLE_0:def 4;
    then hp in Int(BhIU) by A14,TOPS_1:23;
    then consider s be Real such that
A17:  s>0
    and
A18:  Ball(HP,s) c= BhIU by GOBOARD6:5;
    set W=Ball(hp,s);
    reconsider hW=h"W as Subset of M by A8,XBOOLE_1:1;
A19:W c= B /\ (h.:IU) by A18,TOPREAL9:13;
    then reconsider w=W as Subset of TR|cl_Ball(0.TR,1) by XBOOLE_1:1;
A20:w in the topology of TR by PRE_TOPC:def 2;
    hp is Element of REAL n by EUCLID:22;
    then |. hp-hp .|=0;
    then hp in Ball(hp,s) by A17;
    then
A21: p in hW by A8,A11,A15,A6,FUNCT_1:def 7;
    rng h = [#](TR|cl_Ball(0.TR,1)) by A3,TOPS_2:def 5;
    then h.:(h"W) = W by A19,XBOOLE_1:1,FUNCT_1:77;
    then
A22: Tdisk(0.TR,1) | (h.:(h"W)) = TR|W by A9,PRE_TOPC:7;
    w/\cl_Ball(0.TR,1) = w by A9,XBOOLE_1:28;
    then
A23: w in the topology of TR |cl_Ball(0.TR,1) by A9,A20,PRE_TOPC:def 4;
    B /\ (h.:IU) c= h.:IU by XBOOLE_1:17;
    then W c= h.:IU by A19;
    then
A24: hW c= IU by A12,RELAT_1:143;
    reconsider w as open Subset of TR |cl_Ball(0.TR,1) by A23,PRE_TOPC:def 2;
    reconsider hh=h| (h"w) as Function of (M|U) |h"w,
      Tdisk(0.TR,1) | (h.:(h"w)) by JORDAN24:12;
A25: (M|U) |h"W = M|hW by A8,PRE_TOPC:7;
    then reconsider HH=hh as Function of M|hW, TR|W by A22;
    h"w is open by A3,TOPGRP_1:26;
    then hW is open by A24,TSEP_1:9;
    then p in Int hW by A21,TOPS_1:23;
    then reconsider HW=hW as a_neighborhood of p by CONNSP_2:def 1;
    HH is being_homeomorphism by A3,METRIZTS:2,A22,A25;
    then
A27: M|HW, TR|W are_homeomorphic by T_0TOPSP:def 1;
    Tball(hp,s),Tball(0.TR,1) are_homeomorphic by A17,Th3;
    then M|HW, Tball(0.TR,1) are_homeomorphic by A27,A17,BORSUK_3:3;
    then p in Int M by Def4;
    then not p in [#]M\ Int M by XBOOLE_0:def 5;
    hence contradiction by SUBSET_1:def 4,A1;
  end;
  given U be a_neighborhood of p, n be Nat, h be Function of M|U,
    Tdisk(0.TOP-REAL n,1) such that
A28:  h is being_homeomorphism
    and
A29:  h.p in Sphere(0.TOP-REAL n,1);
  set TR=TOP-REAL n;
A30: p in Int U by CONNSP_2:def 1;
  assume not p in Fr M;
  then not p in [#]M\Int M by SUBSET_1:def 4;
  then p in Int M by XBOOLE_0:def 5;
  then consider W be a_neighborhood of p,m such that
A31: M|W,Tball(0.TOP-REAL m,1) are_homeomorphic by Def4;
A33: p in Int W by CONNSP_2:def 1;
  M|U,Tdisk(0.TOP-REAL n,1) are_homeomorphic by A28,T_0TOPSP:def 1;
  then m=n by A30,A33,XBOOLE_0:3,BROUWER3:15,A31;
  then consider g be Function of M|W,TR|Ball(0.TR,1) such that
A34: g is being_homeomorphism by A31,T_0TOPSP:def 1;
A35: Int U c= U by TOPS_1:16;
  set I = (Int U)/\Int W;
A36: [#](M|U)=U by PRE_TOPC:def 5;
  I c= Int U by XBOOLE_1:17;
  then reconsider IU = I as non empty open Subset of (M|U)
    by XBOOLE_1:1,A35,A36,A30,A33,XBOOLE_0:def 4,TSEP_1:9;
A37: dom h = [#](M|U) by A28,TOPS_2:def 5;
  then
A38: h"(h.:IU) = IU by A28,FUNCT_1:94;
  p in I by A30,A33,XBOOLE_0:def 4;
  then
A39: h.p in h.:IU by A37,FUNCT_1:def 6;
  h.:IU is open by A28,TOPGRP_1:25;
  then h.:IU in the topology of TR|cl_Ball(0.TR,1) by PRE_TOPC:def 2;
  then consider Q be Subset of TR such that
A40: Q in the topology of TR
    and
A41: h.:IU = Q /\[#](TR|cl_Ball(0.TR,1)) by PRE_TOPC: def 4;
  reconsider Q as non empty Subset of TR by A41;
A42: Int Q=Q by A40,PRE_TOPC:def 2,TOPS_1:23;
A43: I c= Int W by XBOOLE_1:17;
A44: [#] (TR|cl_Ball(0.TR,1)) = cl_Ball(0.TR,1) by PRE_TOPC :def 5;
  then h.p in cl_Ball(0.TR,1) by A39;
  then reconsider hp=h.p as Point of TR;
  the TopStruct of TR=TopSpaceMetr Euclid n by EUCLID:def 8;
  then reconsider HP=hp as Point of Euclid n by TOPMETR:12;
  hp in Q by A39,A41,XBOOLE_0:def 4;
  then consider s be Real such that
A45: s>0
  and
A46: Ball(HP,s) c= Q by A42,GOBOARD6:5;
  set s2=s/2;
  hp is Element of REAL n by EUCLID:22;
  then |. hp-hp .|=0;
  then
A47:hp in Ball(hp,s2) by A45;
  set clb = cl_Ball(hp,s2)/\cl_Ball(0.TR,1);
A48: Ball(hp,s2) c= cl_Ball(hp,s2) by TOPREAL9:16;
  clb = cl_Ball(hp,s2)/\[#](TR|cl_Ball(0.TR,1)) by PRE_TOPC :def 5;
  then reconsider CLB = clb as non empty Subset of TR|cl_Ball(0.TR,1)
    by A48,A47,A39,XBOOLE_0:def 4;
A49: rng h = [#] (TR|cl_Ball(0.TR,1)) by A28,TOPS_2:def 5;
  then reconsider hCLB=h"CLB as non empty Subset of M|U by RELAT_1:139;
A50:Ball(HP,s)=Ball(hp,s) by TOPREAL9:13;
  hp in CLB by A48,A47,A39,A44,XBOOLE_0:def 4;
  then
A51: p in hCLB by A37,A36,A35,A30,FUNCT_1:def 7;
  n in NAT by ORDINAL1:def 12;
  then
A52: cl_Ball(hp,s2) c= Ball(hp,s) by XREAL_1:216,A45,JORDAN:21;
  then cl_Ball(hp,s2) c= Q by A46,A50;
  then CLB c= h.:IU by A44,XBOOLE_1:26,A41;
  then
A53: hCLB c= IU by RELAT_1:143,A38;
  reconsider BB = Ball(hp,s2)/\[#](TR|cl_Ball(0.TR,1)) as
    Subset of TR|cl_Ball(0.TR,1);
  reconsider hBB =h"BB as Subset of M by A36,XBOOLE_1:1;
  Ball(hp,s2) c= Q by A46,A50,A48,A52;
  then BB c= h.:IU by XBOOLE_1:26,A41;
  then
A54: h"BB c= IU by RELAT_1: 143,A38;
  reconsider HCLB=hCLB as non empty Subset of M by A36,XBOOLE_1:1;
A55: h.:hCLB = CLB by A49,FUNCT_1:77;
A56:(TR|cl_Ball(0.TR,1)) |CLB = TR|clb by A44,PRE_TOPC:7;
A57:(M|U) |hCLB = M|HCLB by A36,PRE_TOPC:7;
  then reconsider h1=h|hCLB as Function of M|HCLB,TR|clb
    by A56,A55,JORDAN24:12;
A58: Int W c= W by TOPS_1:16;
A59: [#](M|W)=W by PRE_TOPC:def 5;
  then reconsider IW = I as non empty open Subset of (M|W)
    by A30,A33,XBOOLE_0:def 4,XBOOLE_1:1,A58,A43,TSEP_1:9;
A60: IU c= W by A58,A43;
  then reconsider hCLW=hCLB as non empty Subset of M|W by A53,XBOOLE_1:1,A59;
A61:(M|W) |hCLW = M|HCLB by A53,A60,XBOOLE_1:1,PRE_TOPC:7;
  set ghCLW = g.:hCLW;
A62: [#] (TR|Ball(0.TR,1)) = Ball(0.TR,1) by PRE_TOPC:def 5;
  then reconsider GhCLW = ghCLW as non empty Subset of TR by XBOOLE_1:1;
A63:(TR|Ball(0.TR,1)) | ghCLW = TR|GhCLW by A62,PRE_TOPC:7;
  then reconsider g1=g|hCLB as Function of M|HCLB,TR|GhCLW by A61,JORDAN24:12;
A64:g1 is being_homeomorphism by A34,METRIZTS:2,A63,A61;
  then
A65:g1" is being_homeomorphism by TOPS_2:56;
A66: dom g = [#](M|W) by A34,TOPS_2:def 5;
  then g.p in GhCLW by A51,FUNCT_1:def 6;
  then reconsider gp=g.p as Point of TR;
  I c= W by A58,A43;
  then reconsider hBW=hBB as Subset of (M|W) by A54,XBOOLE_1:1,A59;
  reconsider ghBW=g.:hBW as Subset of TR by A62,XBOOLE_1:1;
  hp in BB by A47,A39,XBOOLE_0:def 4;
  then p in h"BB by A37,A36,A35,A30,FUNCT_1:def 7;
  then
A67:gp in ghBW by A66,FUNCT_1:def 6;
  set hg= h1*(g1");
  h1 is being_homeomorphism by A28,A56,METRIZTS:2,A57,A55;
  then
A68:hg is being_homeomorphism by A65,TOPS_2:57;
  then
A69: dom hg = [#](TR|GhCLW) by TOPS_2:def 5;
  BB c= clb by TOPREAL9:16,XBOOLE_1:26,A44;
  then
A70: hBB c= hCLB by RELAT_1:143;
  then ghBW c= GhCLW by RELAT_1:123;
  then gp in GhCLW by A67;
  then
A71:gp in dom hg by A69,PRE_TOPC:def 5;
  Ball(hp,s2) in the topology of TR by PRE_TOPC:def 2;
  then BB in the topology of TR|cl_Ball(0.TR,1) by PRE_TOPC:def 4;
  then BB is non empty open by A47,A39,XBOOLE_0:def 4,PRE_TOPC:def 2;
  then h"BB is open by A28,TOPGRP_1:26;
  then hBB is open by A54,TSEP_1:9;
  then hBW is open by TSEP_1:9;
  then g.:hBW is open by A34,TOPGRP_1:25;
  then ghBW is open by A62,TSEP_1:9;
  then gp in Int GhCLW by A67,TOPS_1:22,A70,RELAT_1:123;
  then
A72: hg.gp in Int clb by BROUWER3:11,A68;
  Int clb = (Int cl_Ball(hp,s2)) /\ Int cl_Ball(0.TR,1) by TOPS_1:17;
  then
A73: hg.gp in Int cl_Ball(0.TR,1) by A72,XBOOLE_0:def 4;
  reconsider G1=g1 as Function;
  Fr cl_Ball(0.TR,1) = Sphere(0.TR,1) by BROUWER2:5;
  then
A74:Int cl_Ball(0.TR,1) = cl_Ball(0.TR,1)\Sphere(0.TR,1) by TOPS_1:40;
A75:G1"=g1" by A64,TOPS_2:def 4;
  dom g1 = [#](M|HCLB) by A64,TOPS_2:def 5;
  then p in dom g1 by PRE_TOPC:def 5,A51;
  then
A76: (g1".(g1.p)) = p by A64,A75,FUNCT_1:34;
A77:g.p = g1.p by A51, FUNCT_1:49;
  then
A78: p in dom h1 by A71,FUNCT_1:11,A76;
  hg.gp = h1.p by A71,FUNCT_1:12,A76,A77;
  then hg.gp = h.p by A78,FUNCT_1:47;
  hence contradiction by A29, A73,A74,XBOOLE_0:def 5;
end;
