let n be Nat; :: thesis: for M being non empty TopSpace holds

( ( M is without_boundary & M is n -locally_euclidean ) iff for p being Point of M ex U being a_neighborhood of p ex B being ball Subset of (TOP-REAL n) st U,B are_homeomorphic )

let M be non empty TopSpace; :: thesis: ( ( M is without_boundary & M is n -locally_euclidean ) iff for p being Point of M ex U being a_neighborhood of p ex B being ball Subset of (TOP-REAL n) st U,B are_homeomorphic )

( ( M is without_boundary & M is n -locally_euclidean ) iff for p being Point of M ex U being a_neighborhood of p ex B being ball Subset of (TOP-REAL n) st U,B are_homeomorphic )

let M be non empty TopSpace; :: thesis: ( ( M is without_boundary & M is n -locally_euclidean ) iff for p being Point of M ex U being a_neighborhood of p ex B being ball Subset of (TOP-REAL n) st U,B are_homeomorphic )

hereby :: thesis: ( ( for p being Point of M ex U being a_neighborhood of p ex B being ball Subset of (TOP-REAL n) st U,B are_homeomorphic ) implies ( M is without_boundary & M is n -locally_euclidean ) )

assume A3:
for p being Point of M ex U being a_neighborhood of p ex B being ball Subset of (TOP-REAL n) st U,B are_homeomorphic
; :: thesis: ( M is without_boundary & M is n -locally_euclidean )assume
( M is without_boundary & M is n -locally_euclidean )
; :: thesis: for p being Point of M ex U being a_neighborhood of p ex B being ball Subset of (TOP-REAL n) st U,B are_homeomorphic

then AA: for p being Point of M ex U being a_neighborhood of p ex S being open Subset of (TOP-REAL n) st U,S are_homeomorphic by Def4;

let p be Point of M; :: thesis: ex U being a_neighborhood of p ex B being ball Subset of (TOP-REAL n) st U,B are_homeomorphic

consider U being a_neighborhood of p, B being non empty ball Subset of (TOP-REAL n) such that

A2: U,B are_homeomorphic by AA, Lm1;

reconsider B = B as ball Subset of (TOP-REAL n) ;

take U = U; :: thesis: ex B being ball Subset of (TOP-REAL n) st U,B are_homeomorphic

take B = B; :: thesis: U,B are_homeomorphic

thus U,B are_homeomorphic by A2; :: thesis: verum

end;then AA: for p being Point of M ex U being a_neighborhood of p ex S being open Subset of (TOP-REAL n) st U,S are_homeomorphic by Def4;

let p be Point of M; :: thesis: ex U being a_neighborhood of p ex B being ball Subset of (TOP-REAL n) st U,B are_homeomorphic

consider U being a_neighborhood of p, B being non empty ball Subset of (TOP-REAL n) such that

A2: U,B are_homeomorphic by AA, Lm1;

reconsider B = B as ball Subset of (TOP-REAL n) ;

take U = U; :: thesis: ex B being ball Subset of (TOP-REAL n) st U,B are_homeomorphic

take B = B; :: thesis: U,B are_homeomorphic

thus U,B are_homeomorphic by A2; :: thesis: verum

now :: thesis: for p being Point of M ex U being a_neighborhood of p ex S being open Subset of (TOP-REAL n) st U,S are_homeomorphic

hence
( M is without_boundary & M is n -locally_euclidean )
by Def4; :: thesis: verumlet p be Point of M; :: thesis: ex U being a_neighborhood of p ex S being open Subset of (TOP-REAL n) st U,S are_homeomorphic

consider U being a_neighborhood of p, B being ball Subset of (TOP-REAL n) such that

A4: U,B are_homeomorphic by A3;

reconsider S = B as open Subset of (TOP-REAL n) ;

take U = U; :: thesis: ex S being open Subset of (TOP-REAL n) st U,S are_homeomorphic

take S = S; :: thesis: U,S are_homeomorphic

thus U,S are_homeomorphic by A4; :: thesis: verum

end;consider U being a_neighborhood of p, B being ball Subset of (TOP-REAL n) such that

A4: U,B are_homeomorphic by A3;

reconsider S = B as open Subset of (TOP-REAL n) ;

take U = U; :: thesis: ex S being open Subset of (TOP-REAL n) st U,S are_homeomorphic

take S = S; :: thesis: U,S are_homeomorphic

thus U,S are_homeomorphic by A4; :: thesis: verum