
theorem
  for S, T being TopSpace, A being Subset of S, B being Subset of T st A
  is connected & B is connected holds [:A,B:] is connected
proof
  let S, T be TopSpace;
  let A be Subset of S;
  let B be Subset of T;
  assume S|A is connected & T|B is connected;
  then reconsider SA = S|A, TB = T|B as connected TopSpace;
  [:SA,TB:] is connected;
  hence [:S,T:] | [:A,B:] is connected by BORSUK_3:22;
end;
