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

theorem Th11:
  for M, N being non empty TopSpace
  for p being Point of M, U being a_neighborhood of p, B being open Subset of N
  st U,B are_homeomorphic
  ex V being open Subset of M, S being open Subset of N
     st V c= U & p in V & V,S are_homeomorphic
proof
  let M, N be non empty TopSpace;
  let p be Point of M;
  let U be a_neighborhood of p;
  let B be open Subset of N;
  assume A0:U,B are_homeomorphic; then
  A1: M|U, N|B are_homeomorphic by METRIZTS:def 1;:: then
  consider f be Function of M|U, N|B such that
  A2: f is being_homeomorphism by T_0TOPSP:def 1,A0, METRIZTS:def 1;
  consider V be Subset of M such that
  A3: V is open & V c= U & p in V by CONNSP_2:6;
  V c= [#](M|U) by A3,PRE_TOPC:def 5; then
  reconsider V1 = V as Subset of M|U;
  reconsider M1=M|U as non empty TopStruct by A3;
  reconsider N1=N|B as non empty TopStruct by A3,A1,YELLOW14:18;
  reconsider f1=f as Function of M1, N1;
  rng f c= [#](N|B); then
  A4: rng f c= B by PRE_TOPC:def 5;
::  V1, f .: V1 are_homeomorphic by A2,METRIZTS:3; then
  A5: (M|U) | V1 , (N|B) | (f .: V1) are_homeomorphic
     by METRIZTS:def 1,A2,METRIZTS:3;
  reconsider V as open Subset of M by A3;
  B = the carrier of N|B by PRE_TOPC:8; then
  reconsider N1 = N|B as open SubSpace of N by TSEP_1:16;
  B c= the carrier of N; then
  [#](N|B) c= the carrier of N by PRE_TOPC:def 5; then
  reconsider S = f .: V1 as Subset of N by XBOOLE_1:1;
  reconsider S1 = f .: V1 as Subset of N1;
  A6: for P being Subset of M1 holds P is open iff f1 .: P is open
  by A2,TOPGRP_1:25;
  A7: V in the topology of M by PRE_TOPC:def 2;
  V1 = V /\ [#]M1 by XBOOLE_1:28; then
  V1 in the topology of M1 by A7,PRE_TOPC:def 4; then
  ::V1 is open by PRE_TOPC:def 2; then
  S1 is open by A6,PRE_TOPC:def 2; then
  reconsider S as open Subset of N by TSEP_1:17;
  take V, S;
  thus V c= U & p in V by A3;
  A8: (M | U) | V1 = M | V by A3,PRE_TOPC:7;
  f .: U c= rng f by RELAT_1:111; then
  A9: f .: U c= B by A4;
  f .: V c= f .: U by A3,RELAT_1:123; then
  (N | B) | (f .: V1) = N | S by A9,PRE_TOPC:7,XBOOLE_1:1;
  hence V,S are_homeomorphic by A5,A8,METRIZTS:def 1;
end;
