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

let 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