
theorem
  for X, Y being non empty TopSpace, x being Point of X, f being
  Function of [:Y, X | {x}:], Y st f = pr1(the carrier of Y, {x}) holds f is
  being_homeomorphism
proof
  let X, Y be non empty TopSpace, x be Point of X, f be Function of [:Y, X | {
  x}:], Y;
  set Z = {x};
  assume
A1: f = pr1(the carrier of Y, Z);
  thus dom f = [#][:Y, (X|Z):] by FUNCT_2:def 1;
  thus rng f = [#]Y by A1,FUNCT_3:44;
  thus f is one-to-one by A1,Th4;
  the carrier of (X|Z) = Z by PRE_TOPC:8;
  hence f is continuous by A1,YELLOW12:39;
  reconsider Z as non empty Subset of X;
  reconsider idZ = Y --> x as continuous Function of Y, (X|Z) by Th2;
  reconsider KA = <:id Y, idZ:> as continuous Function of Y, [:Y, (X|Z):] by
YELLOW12:41;
  KA = f" by A1,Th6;
  hence thesis;
end;
