
theorem Th69:
  for I being 2-element set
  for J being TopSpace-yielding non-Empty ManySortedSet of I
  for i,j being Element of I, P being Subset of product Carrier J st i <> j
  holds P in product_prebasis J iff
    (ex V being Subset of J.i st
      V is open & P = product((i,j) --> (V,[#](J.j)) ) ) or
    (ex W being Subset of J.j st
      W is open & P = product((i,j) --> ([#](J.i),W) ) )
proof
  let I be 2-element set;
  let J be TopSpace-yielding non-Empty ManySortedSet of I;
  let i, j be Element of I, P be Subset of product Carrier J;
  assume A1: i <> j;
  hence P in product_prebasis J implies
    (ex V being Subset of J.i st
      V is open & P = product( (i,j) --> (V,[#](J.j)) ) ) or
    (ex W being Subset of J.j st
      W is open & P = product( (i,j) --> ([#](J.i),W) ) ) by Lm8;
  assume
    (ex V being Subset of J.i st
      V is open & P = product( (i,j) --> (V,[#](J.j)) ) ) or
    (ex W being Subset of J.j st
      W is open & P = product( (i,j) --> ([#](J.i),W) ) );
  then per cases;
  suppose ex V being Subset of J.i st
    V is open & P = product( (i,j) --> (V,[#](J.j)) );
    then consider V being Subset of J.i such that
      A2: V is open & P = product( (i,j) --> (V,[#](J.j)) );
      (Carrier J).j = [#](J.j) by PENCIL_3:7;
      then P = product ((Carrier J) +* (i,V)) by A1, A2, Th34;
    hence P in product_prebasis J by A2,WAYBEL18:def 2;
  end;
  suppose ex W being Subset of J.j st
    W is open & P = product( (i,j) --> ([#](J.i),W) );
    then consider W being Subset of J.j such that
      A3: W is open & P = product( (i,j) --> ([#](J.i),W) );
      (Carrier J).i = [#](J.i) by PENCIL_3:7;
      then P = product ((Carrier J) +* (j,W)) by A1, A3, Th34;
    hence P in product_prebasis J by A3,WAYBEL18:def 2;
  end;
end;
