theorem Th120:
  for f being Function of X,Y, X1, X2 being non empty SubSpace of
  X st X = X1 union X2 & X1,X2 are_weakly_separated 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 non empty SubSpace of X such that
A1: X = X1 union X2 and
A2: X1,X2 are_weakly_separated;
  thus f is continuous Function of X,Y implies f|X1 is continuous Function of
  X1,Y & f|X2 is continuous Function of X2,Y;
  assume
  f|X1 is continuous Function of X1,Y & f|X2 is continuous Function of X2,Y;
  then f|(X1 union X2) is continuous Function of X1 union X2,Y by A2,Th117;
  hence thesis by A1,Th54;
end;
