reserve e,u for set;
reserve X, Y for non empty TopSpace;

theorem Th6:
  for X, Y being TopSpace for V being Subset of X, W being Subset
  of Y st V is open & W is open holds [:V,W:] is open
proof
  let X, Y be TopSpace, V be Subset of X, W be Subset of Y such that
A1: V is open & W is open;
  reconsider PP = {[:V,W:]} as Subset-Family of [:X,Y:];
  reconsider PP as Subset-Family of [:X,Y:];
A2: now
    let e;
    assume
A3: e in PP;
    take V,W;
    thus e = [:V,W:] & V is open & W is open by A1,A3,TARSKI:def 1;
  end;
  [:V,W:] = union {[:V,W:]};
  hence thesis by A2,Th5;
end;
