let n be natural number ; :: thesis: for M being non empty TopSpace holds
( M is n -locally_euclidean iff for p being Point of M ex U being a_neighborhood of p st U, [#] (TOP-REAL n) are_homeomorphic )
let M be non empty TopSpace; :: thesis: ( M is n -locally_euclidean iff for p being Point of M ex U being a_neighborhood of p st U, [#] (TOP-REAL n) are_homeomorphic )

hereby :: thesis: ( ( for p being Point of M ex U being a_neighborhood of p st U, [#] (TOP-REAL n) are_homeomorphic ) implies M is n -locally_euclidean )

assume A1: M is n -locally_euclidean ; :: thesis: for p being Point of M ex U being a_neighborhood of p st U, [#] (TOP-REAL n) are_homeomorphic
let p be Point of M; :: thesis: ex U being a_neighborhood of p st U, [#] (TOP-REAL n) 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 A1, Lm1;
take U = U; :: thesis: U, [#] (TOP-REAL n) are_homeomorphic
A3: (TOP-REAL n) | ([#] (TOP-REAL n)) = TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) by TSEP_1:93;
reconsider B1 = (TOP-REAL n) | B as non empty TopSpace ;
M | U,B1 are_homeomorphic by A2, METRIZTS:def 1;
then reconsider U1 = M | U as non empty TopSpace by YELLOW14:18;
A4: U1,B1 are_homeomorphic by A2, METRIZTS:def 1;
B, [#] (TOP-REAL n) are_homeomorphic by Th10;
then A5: B1, TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) are_homeomorphic by A3, METRIZTS:def 1;
U1, TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) are_homeomorphic by A4, A5, BORSUK_3:3;
hence U, [#] (TOP-REAL n) are_homeomorphic by A3, METRIZTS:def 1; :: thesis: verum

end;

assume A6: for p being Point of M ex U being a_neighborhood of p st U, [#] (TOP-REAL n) are_homeomorphic ; :: thesis: M is n -locally_euclidean

now

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 such that
A7: U, [#] (TOP-REAL n) are_homeomorphic by A6;
set S = the non empty ball Subset of (TOP-REAL n);
A8: the non empty ball Subset of (TOP-REAL n), [#] (TOP-REAL n) are_homeomorphic by Th10;
reconsider S = the non empty ball Subset of (TOP-REAL n) 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
A9: (TOP-REAL n) | ([#] (TOP-REAL n)) = TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) by TSEP_1:93;
A10: M | U, TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) are_homeomorphic by A7, A9, METRIZTS:def 1;
then reconsider U1 = M | U as non empty TopSpace by YELLOW14:18;
reconsider S1 = (TOP-REAL n) | S as non empty TopSpace ;
A11: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #),S1 are_homeomorphic by A8, A9, METRIZTS:def 1;
U1,S1 are_homeomorphic by A10, A11, BORSUK_3:3;
hence U,S are_homeomorphic by METRIZTS:def 1; :: thesis: verum

end;

hence M is n -locally_euclidean by Def4; :: thesis: verum