
theorem Th67:
  for I being 1-element set
  for J being TopSpace-yielding non-Empty ManySortedSet of I
  for i being Element of I holds proj(J,i) is being_homeomorphism
proof
  let I be 1-element set;
  let J be TopSpace-yielding non-Empty ManySortedSet of I;
  let i be Element of I;
  card I = 1 by CARD_1:def 7;
  then A1: I = {i} by ORDERS_5:2;
  :: we already have all properties ready and just need to collect them
  set f = proj(J,i);
  A2: dom f = the carrier of product J by FUNCT_2:def 1
    .= [#]product J by STRUCT_0:def 3;
  the carrier of J.i = [#](J.i) by STRUCT_0:def 3
    .= (Carrier J).i by PENCIL_3:7;
  then A3: Carrier J = {i} --> the carrier of J.i by A1, Th7;
  A4: rng f = the carrier of J.i by FUNCT_2:def 3
    .= [#](J.i) by STRUCT_0:def 3;
  a5: f = proj(Carrier J,i) by WAYBEL18:def 4
    .= proj(i .--> the carrier of J.i,i) by A3, FUNCOP_1:def 9;
  f" is continuous by a5, TOPREALA:14;
  hence f is being_homeomorphism by A2, A4, a5, TOPS_2:def 5;
end;
