
theorem
  for T1,T2,S1,S2 being non empty TopSpace
  for f being Function of T1, S1, g being Function of T2, S2
  st f is open & g is open holds [: f,g :] is open
proof
  let T1,T2,S1,S2 be non empty TopSpace;
  let f be Function of T1, S1, g be Function of T2, S2;
  assume A1: f is open & g is open;
  ex B being Basis of [: T1,T2 :] st
    for P being Subset of [: T1, T2 :] st P in B holds [: f,g :].:P is open
  proof
    set B1 = the Basis of T1;
    set B2 = the Basis of T2;
    set B = {[: V,W :] where V is Subset of T1, W is Subset of T2 :
      V in B1 & W in B2};
    reconsider B as Basis of [: T1, T2 :] by YELLOW_9:40;
    take B;
    let P be Subset of [: T1, T2 :];
    assume P in B;
    then consider V being Subset of T1, W being Subset of T2 such that
      A2: P = [: V,W :] & V in B1 & W in B2;
    A3: f.:V is open & g.:W is open by A1, T_0TOPSP:def 2,A2, TOPS_2:def 1;
    [: f,g :].:P = [: f.:V,g.:W :] by A2, FUNCT_3:72;
    hence thesis by A3, BORSUK_1:6;
  end;
  hence thesis by Th45;
end;
