theorem
  for f being Function of X,Y, X1, X2 being closed non empty SubSpace of
  X holds f|(X1 union X2) is continuous Function of X1 union X2,Y iff f|X1 is
  continuous Function of X1,Y & f|X2 is continuous Function of X2,Y
proof
  let f be Function of X,Y, X1, X2 be closed non empty SubSpace of X;
  X1,X2 are_weakly_separated by TSEP_1:80;
  hence thesis by Th117;
end;
