
theorem Th16:
  for I being non empty set, J being TopStruct-yielding non-Empty
  ManySortedSet of I, A being set st A in product_prebasis J ex i being Element
  of I, Ai being Subset of J.i st Ai is open & proj(J,i)"Ai = A
proof
  let I be non empty set, J be TopStruct-yielding non-Empty ManySortedSet of I,
  A be set;
  assume A in product_prebasis J;
  then consider i being set, T being TopStruct, Ai being Subset of T such that
A1: i in I and
A2: Ai is open and
A3: T = J.i and
A4: A = product ((Carrier J) +* (i,Ai)) by WAYBEL18:def 2;
  reconsider i as Element of I by A1;
  reconsider Ai as Subset of J.i by A3;
  take i;
  take Ai;
  thus Ai is open by A2,A3;
  thus thesis by A4,WAYBEL18:4;
end;
