
theorem Th13:
  for X, Y being non empty TopSpace, x being Point of X holds [: X
  | {x}, Y :], Y are_homeomorphic
proof
  let X be non empty TopSpace, Y be non empty TopSpace, x be Point of X;
  set Z = {x};
  the carrier of [:(X|Z), Y:] = [:the carrier of (X|Z), the carrier of Y
  :] by BORSUK_1:def 2
    .= [:Z, the carrier of Y:] by PRE_TOPC:8;
  then reconsider f= pr2(Z, the carrier of Y) as Function of [:X|Z, Y:], Y;
  take f;
  thus thesis by Th9;
end;
