reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;
reserve f for Function of X,Y;
reserve X,Y,Z for non empty TopSpace;
reserve f for Function of X,Y,
  g for Function of Y,Z;
reserve X, Y for non empty TopSpace,
  X0 for non empty SubSpace of X;
reserve f for Function of X,Y;
reserve f for Function of X,Y,
  X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace,
  X0, X1 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X0, X1, X2 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X for non empty TopSpace,
  H, G for Subset of X;
reserve A for Subset of X;
reserve X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;

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;
