
theorem Th9:
  for X, Y being non empty TopSpace, x being Point of X, f being
  Function of [:X | {x}, Y:], Y st f = pr2({x}, the carrier of Y) holds f is
  being_homeomorphism
proof
  let X, Y be non empty TopSpace, x be Point of X, f be Function of [:X | {x},
  Y:], Y;
  set Z = {x};
  assume
A1: f = pr2(Z, the carrier of Y);
  thus dom f = [#][:(X|Z), Y:] by FUNCT_2:def 1;
  thus rng f = [#]Y by A1,FUNCT_3:46;
  thus f is one-to-one by A1,Th5;
  the carrier of (X|Z) = Z by PRE_TOPC:8;
  hence f is continuous by A1,YELLOW12:40;
  reconsider Z as non empty Subset of X;
  reconsider idZ = Y --> x as continuous Function of Y, (X|Z) by Th2;
  reconsider KA = <:idZ, id Y:> as continuous Function of Y, [:(X|Z), Y:] by
YELLOW12:41;
  KA = f" by A1,Th7;
  hence thesis;
end;
