theorem Th121:
  for f being Function of X,Y, X1, X2 being closed non empty
SubSpace of X st X = X1 union X2 holds f is continuous Function of X,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 such that
A1: X = X1 union X2;
  X1,X2 are_weakly_separated by TSEP_1:80;
  hence thesis by A1,Th120;
end;
