reserve T,T1,T2,S for non empty TopSpace;
reserve GY for non empty TopSpace,
  r,s for Real;

theorem Th8:
  for S1, S2, T1, T2 be non empty TopSpace, f be continuous
Function of S1, T1, g be continuous Function of S2, T2, P1, P2 being Subset of
  [:T1, T2:] holds (P2 in Base-Appr P1 implies [:f,g:]"P2 is open)
proof
  let S1, S2, T1, T2 be non empty TopSpace, f be continuous Function of S1, T1
  , g be continuous Function of S2, T2, P1, P2 be Subset of [:T1, T2:];
  assume P2 in Base-Appr P1;
  then consider X11 be Subset of T1, Y11 be Subset of T2 such that
A1: P2 = [:X11,Y11:] and
  [:X11,Y11:] c= P1 and
A2: X11 is open and
A3: Y11 is open;
  [#]T1 <> {};
  then
A4: f"X11 is open by A2,TOPS_2:43;
  [#]T2 <> {};
  then
A5: g"Y11 is open by A3,TOPS_2:43;
  [:f,g:]"P2 = [:f"X11, g"Y11:] by A1,FUNCT_3:73;
  hence thesis by A4,A5,BORSUK_1:6;
end;
