 reserve n,m for Nat,
         p,q for Point of TOP-REAL n, r for Real;

theorem
  for M be non empty TopSpace
    for q be Point of M,r be real number,p be Point of TOP-REAL n st r>0
    for U be a_neighborhood of q st M|U,Tball(p,r) are_homeomorphic
    ex W being a_neighborhood of q st
      W c= Int U & M|W,Tdisk(p,r) are_homeomorphic
 proof
   let M be non empty TopSpace;
   set TR=TOP-REAL n;
   let q be Point of M,r be real number,p be Point of TR such that
A1: r>0;
   let U be a_neighborhood of q such that
A2:  M|U,Tball(p,r) are_homeomorphic;
A3:  [#](M|U)=U by PRE_TOPC:def 5;
   then reconsider IU=Int U as Subset of M|U by TOPS_1:16;
   set MU=M|U;
   consider h be Function of MU,Tball(p,r) such that
A4:  h is being_homeomorphism by T_0TOPSP:def 1,A2;
A5:dom h =[#](M|U) by A4,TOPS_2:def 5;
A7: [#]Tball(p,r) = Ball(p,r) by PRE_TOPC:def 5;
   then reconsider W=h.:IU as Subset of TR by XBOOLE_1:1;
   IU is open by TSEP_1:9;
   then h .: IU is open by A4,TOPGRP_1:25,A1;
   then reconsider W as open Subset of TR by A7,TSEP_1:9;
   q in Int U by CONNSP_2:def 1;
   then
A8:  h.q in W by A5,FUNCT_1:def 6;
   then reconsider hq=h.q as Point of TR;
   reconsider HQ=hq as Point of Euclid n by EUCLID:67;
   Int W=W by TOPS_1:23;
   then consider s be Real such that
A9: s>0
   and
A10: Ball(HQ,s) c= W by A8,GOBOARD6:5;
   set m=s/2;
A11: Ball(HQ,s)=Ball(hq,s) by TOPREAL9:13;
   set CL=cl_Ball(hq,m);
A12: n in NAT by ORDINAL1:def 12;
A13: CL c= Ball(hq,s) by A9,A12,XREAL_1:216,JORDAN:21;
   then CL c= W by A10,A11;
   then
A14: CL c= Ball(p,r) by A7,XBOOLE_1:1;
   set BB = Ball(hq,m);
A15: BB c= CL by TOPREAL9:16;
   then reconsider CL,BB as Subset of Tball(p,r) by A14,XBOOLE_1:1,A7;
A16: rng h = [#]Tball(p,r) by A4,TOPS_2:def 5;
A17: q in Int U by CONNSP_2:def 1;
A18: Int U c= U by TOPS_1:16;
   reconsider hBB=h"BB as Subset of M by A3,XBOOLE_1:1;
   hq is Element of REAL n by EUCLID:22;
   then |. hq-hq .|=0;
   then hq in BB by A9;
   then
A19:q in hBB by FUNCT_1:def 7,A5,A18,A17,A3;
A20:dom h = [#]MU by A4,TOPS_2:def 5;
A21:h"W = IU by FUNCT_1:82,A4,FUNCT_1:76,A20;
   CL meets Ball(p,r) by A9,A7,XBOOLE_1:67;
   then reconsider hCL=h"CL as non empty Subset of MU by A7,RELAT_1:138,A16;
A22: h.:hCL = CL by FUNCT_1:77,A16;
A23: Tball(p,r) | CL = TR|cl_Ball(hq,m) by A7,PRE_TOPC:7;
   then reconsider HH=h|hCL as Function of MU|hCL,Tdisk(hq,m)
     by A22,A1,JORDAN24:12;
   HH is being_homeomorphism by A22,A23,A4,METRIZTS:2;
   then
A24: MU|hCL,Tdisk(hq,m) are_homeomorphic by T_0TOPSP:def 1;
   Tdisk(hq,m),Tdisk(p,r) are_homeomorphic by A1,A9,Lm1;
   then
A25: MU|hCL,Tdisk(p,r) are_homeomorphic by A1,A9,BORSUK_3:3,A24;
   reconsider HCL=hCL as Subset of M by A3,XBOOLE_1:1;
A26: MU|hCL=M|HCL by PRE_TOPC:7,A3;
   BB c= W by A10,A11,A13,A15;
   then
A27: h"BB c= h"W by RELAT_1:143;
   BB is open by TSEP_1:9;
   then h"BB is open by A4,TOPGRP_1:26,A1;
   then hBB is open by A21,A27,TSEP_1:9;
   then q in Int HCL by RELAT_1:143,A15,A19,TOPS_1:22;
   then
A28: HCL is a_neighborhood of q by CONNSP_2:def 1;
   CL c= W by A13,A10,A11;
   hence thesis by A28,RELAT_1:143,A21,A25,A26;
 end;
