reserve n for Nat;
reserve p for Point of TOP-REAL n, r for Real;
reserve q for Point of TOP-REAL n;
reserve M for non empty TopSpace;

theorem Def4:
  M is n-locally_euclidean without_boundary iff
    for p being Point of M holds
      ex U being a_neighborhood of p, S being open Subset of TOP-REAL n
    st U,S are_homeomorphic
proof
set TRn=TOP-REAL n;
   hereby assume A1:M is n-locally_euclidean without_boundary;
    let p be Point of M;
    the carrier of M = Int M by A1,MFOLD_0:def 6;
  ::  then p in Int M;
    then consider U be a_neighborhood of p,m be Nat such that 
         A2: M|U,Tball(0.TOP-REAL m,1) are_homeomorphic
        by MFOLD_0:def 4;
    set TR=TOP-REAL m;    
    consider W be a_neighborhood of p such that 
         A3: M|W,Tdisk(0.TRn,1) are_homeomorphic
             by A1,MFOLD_0:def 3;
    p in Int U & p in Int W by CONNSP_2:def 1;
    then n=m by A2,A3,BROUWER3:15,XBOOLE_0:3;
    hence  ex U being a_neighborhood of p, S being open Subset of TRn
    st U,S are_homeomorphic by A2,METRIZTS:def 1;
end;
assume    R: for p being Point of M holds
      ex U being a_neighborhood of p, S being open Subset of TRn
    st U,S are_homeomorphic;
K: for p be Point of M
     ex U be a_neighborhood of p st
       M|U,Tball(0.TRn,1) are_homeomorphic
proof
   let p be Point of M;
   consider U be a_neighborhood of p, B be non empty ball Subset of TRn
     such that
      B1: U,B are_homeomorphic by R,Lm1;
take U;
    consider q be Point of TRn, r be Real such that
       B2: B = Ball(q,r) by Def1;

B3:M|U,TRn|Ball(q,r) are_homeomorphic by B1,B2,METRIZTS:def 1;
p in Int U by CONNSP_2:def 1;
then B4:ex W be Subset of M st W is open & W c= U & p in W by TOPS_1:22;
r>0 by B2;then
    Tball(q,r),Tball(0.TRn,1) are_homeomorphic by MFOLD_0:3;
   hence M|U,Tball(0.TRn,1)  are_homeomorphic by B3,BORSUK_3:3,B4,B2;
end;
ZZ: now let p be Point of M;
   ex U be a_neighborhood of p st
       M|U,Tball(0.TOP-REAL n,1) are_homeomorphic by K;
hence ex U be a_neighborhood of p,n st
       M|U,Tball(0.TOP-REAL n,1) are_homeomorphic;
end;
then  F:M is without_boundary locally_euclidean by MFOLD_0:6;


 for p being Point of M
      ex U being a_neighborhood of p st
        M|U,Tdisk(0.TOP-REAL n,1) are_homeomorphic
proof
   let p be Point of M;
 consider U be a_neighborhood of p,m be Nat such that
    B1:    M|U,Tdisk(0.TOP-REAL m,1) are_homeomorphic by MFOLD_0:def 2,F;

    consider W be a_neighborhood of p such that 
         A3: M|W,Tball(0.TRn,1) are_homeomorphic
             by K;
    p in Int U & p in Int W by CONNSP_2:def 1;
::    then Int U meets Int W by XBOOLE_0:3;
    then n=m by B1,A3,BROUWER3:15,XBOOLE_0:3;
    hence thesis by B1;
end;
::then M is n-locally_euclidean by MFOLD_0:def 3;
hence thesis by ZZ,MFOLD_0:6,def 3;
end;
