reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th12:
  for X being TopSpace, Y being non empty TopSpace for A being
  open closed Subset of X for f being continuous Function of X|A, Y for g being
  continuous Function of X|A`, Y holds f+*g is continuous Function of X, Y
proof
  let X be TopSpace;
  let Y be non empty TopSpace;
  let A be open closed Subset of X;
  let f be continuous Function of X|A, Y;
  let g be continuous Function of X|A`, Y;
  A\/A` = [#]X by PRE_TOPC:2;
  then
A1: X|(A\/A`) = the TopStruct of X by TSEP_1:93;
  A misses A` by XBOOLE_1:79;
  then
A2: f+*g is continuous Function of X|(A\/A`), Y by Th11;
  the TopStruct of Y = the TopStruct of Y;
  hence thesis by A2,A1,YELLOW12:36;
end;
