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

theorem Th9:
  for S1, S2, T1, T2 be non empty TopSpace, f be continuous
Function of S1, T1, g be continuous Function of S2, T2, P2 being Subset of [:T1
  , T2:] holds (P2 is open 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, P2 be Subset of [:T1, T2:];
  reconsider Kill = [:f,g:]"Base-Appr P2 as Subset-Family of [:S1, S2:];
  for P being Subset of [:S1, S2:] holds P in Kill implies P is open
  proof
    let P be Subset of [:S1, S2:];
    assume P in Kill;
    then
    ex B being Subset of [:T1, T2:] st B in Base-Appr P2 & P = [:f,g:]"B by
FUNCT_2:def 9;
    hence thesis by Th8;
  end;
  then
A1: Kill is open by TOPS_2:def 1;
  assume P2 is open;
  then P2 = union Base-Appr P2 by BORSUK_1:13;
  then
  [:f,g:]"(P2 qua Subset of [:T1, T2:]) = union ([:f,g:]"Base-Appr P2) by Th7;
  hence thesis by A1,TOPS_2:19;
end;
