reserve r for Real;
reserve a, b for Real;
reserve T for non empty TopSpace;
reserve A for non empty SubSpace of T;
reserve P,Q for Subset of T,
  p for Point of T;

theorem
  for P being non empty Subset of T st p in P for Q being a_neighborhood
  of p holds for p9 being Point of T|P, Q9 being Subset of T|P st Q9 = Q /\ P &
  p9= p holds Q9 is a_neighborhood of p9
proof
  let P be non empty Subset of T;
  assume
A1: p in P;
  let Q be a_neighborhood of p;
  let p9 be Point of T|P, Q9 be Subset of T|P such that
A2: Q9 = Q /\ P and
A3: p9= p;
  p in Int Q by CONNSP_2:def 1;
  then consider S being Subset of T such that
A4: S is open and
A5: S c= Q and
A6: p in S by TOPS_1:22;
  reconsider R = S /\ P as Subset of T|P by TOPS_2:29;
A7: R c= Q9 by A5,A2,XBOOLE_1:26;
  S /\ [#](T|P) = S /\ P by PRE_TOPC:def 5;
  then
A8: R is open by A4,TOPS_2:24;
  p9 in R by A1,A6,A3,XBOOLE_0:def 4;
  then p9 in Int Q9 by A8,A7,TOPS_1:22;
  hence thesis by CONNSP_2:def 1;
end;
