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

theorem Th19:
  for H being Subset-Family of [:X,Y:] st H is open holds Pr1(X,Y)
  .:H is open & Pr2(X,Y).:H is open
proof
  let H be Subset-Family of [:X,Y:];
  reconsider P1 = Pr1(X,Y).:H as Subset-Family of X;
  reconsider P2 = Pr2(X,Y).:H as Subset-Family of Y;
  assume
A1: H is open;
A2: P2 is open
  proof
    let Y1 be Subset of Y;
    assume Y1 in P2;
    then consider D being Subset of [:X,Y:] such that
A3: D in H and
A4: Y1 = pr2(the carrier of X, the carrier of Y).:D by Th17;
    D is open by A1,A3;
    hence thesis by A4,Th18;
  end;
  P1 is open
  proof
    let X1 be Subset of X;
    assume X1 in P1;
    then consider D being Subset of [:X,Y:] such that
A5: D in H and
A6: X1 = pr1(the carrier of X, the carrier of Y).:D by Th16;
    D is open by A1,A5;
    hence thesis by A6,Th18;
  end;
  hence thesis by A2;
end;
