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

theorem Th11:
  for X being TopSpace, Y being non empty TopSpace for A,B being
closed Subset of X st A misses B for f being continuous Function of X|A, Y for
g being continuous Function of X|B, Y holds f+*g is continuous Function of X|(A
  \/ B), Y
proof
  let X be TopSpace, Y be non empty TopSpace;
  let A,B be closed Subset of X such that
A1: A misses B;
  let f be continuous Function of X|A, Y;
  let g be continuous Function of X|B, Y;
  the carrier of X|B = B by PRE_TOPC:8;
  then
A2: dom g = B by FUNCT_2:def 1;
  the carrier of X|A = A by PRE_TOPC:8;
  then dom f = A by FUNCT_2:def 1;
  hence thesis by A2,A1,Th10,PARTFUN1:56;
end;
