
theorem Th64:
  for I being 1-element set
  for J being TopStruct-yielding non-Empty ManySortedSet of I
  for i being Element of I, P being Subset of product Carrier J
  holds P in product_prebasis J iff ex V being Subset of J.i
    st V is open & P = product ({i} --> V)
proof
  let I be 1-element set;
  let J be TopStruct-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;
  the carrier of J.i = [#](J.i) by STRUCT_0:def 3
    .= (Carrier J).i by PENCIL_3:7;
  then A2: Carrier J = {i} --> the carrier of J.i by A1, Th7;
  let P be Subset of product Carrier J;
  hereby
    assume P in product_prebasis J;
    then consider j being set, T being TopStruct, V being Subset of T such that
      A3: j in I & V is open & T = J.j and
      A4: P = product ((Carrier J) +* (j,V)) by WAYBEL18:def 2;
    A5: i = j by A1, A3, TARSKI:def 1;
    reconsider W=V as Subset of J.i by A1, A3, TARSKI:def 1;
    take W;
    thus W is open by A3, A5;
    thus P = product ( (i .--> the carrier of J.i) +* (i,W))
        by A2, A4, A5, FUNCOP_1:def 9
      .= product (i .--> W) by Th44
      .= product ({i} --> W) by FUNCOP_1:def 9;
  end;
  given V being Subset of J.i such that
    A6: V is open & P = product ({i} --> V);
    P = product (i .--> V) by A6, FUNCOP_1:def 9
      .= product ( (i .--> the carrier of J.i) +* (i,V)) by Th44
      .= product ((Carrier J) +* (i,V)) by A2, FUNCOP_1:def 9;
  hence thesis by A6,WAYBEL18:def 2;
end;
