reserve X for non empty TopSpace;
reserve x for Point of X;
reserve U1 for Subset of X;

theorem Th9:
  for B being non empty Subset of X, x being Point of X|B, A being
  Subset of X|B, A1 being Subset of X, x1 being Point of X st B is open & A is
  a_neighborhood of x & A = A1 & x = x1 holds A1 is a_neighborhood of x1
proof
  let B be non empty Subset of X, x be Point of X|B, A be Subset of X|B, A1 be
  Subset of X, x1 be Point of X such that
A1: B is open and
A2: A is a_neighborhood of x and
A3: A=A1 & x=x1;
  x in Int A by A2,Def1;
  then consider Q1 being Subset of X|B such that
A4: Q1 is open and
A5: Q1 c= A & x in Q1 by TOPS_1:22;
  Q1 in the topology of X|B by A4,PRE_TOPC:def 2;
  then consider Q being Subset of X such that
A6: Q in the topology of X and
A7: Q1 = Q /\ [#](X|B) by PRE_TOPC:def 4;
  reconsider Q2 = Q as Subset of X;
  Q2 is open by A6,PRE_TOPC:def 2;
  then
A8: Q /\ B is open by A1;
  Q1 = Q /\ B by A7,PRE_TOPC:def 5;
  then x1 in Int A1 by A3,A5,A8,TOPS_1:22;
  hence thesis by Def1;
end;
