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

theorem Th10:
  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 A1 is
  a_neighborhood of x1 & A=A1 & x=x1 holds A is a_neighborhood of x
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: A1 is a_neighborhood of x1 and
A2: A=A1 and
A3: x=x1;
  x1 in Int A1 by A1,Def1;
  then
A4: x1 in Int (A1) /\ [#](X|B) by A3,XBOOLE_0:def 4;
  Int (A1) /\ [#](X|B) c= Int A by A2,Lm1;
  hence thesis by A3,A4,Def1;
end;
