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;

theorem Th63:
  for X,Y,Z being non empty TopSpace, X0 being non empty SubSpace
of X, g being Function of Y,Z, f being Function of X,Y st g is continuous & f|
  X0 is continuous holds (g*f)|X0 is continuous
proof
  let X,Y,Z be non empty TopSpace, X0 be non empty SubSpace of X, g be
  Function of Y,Z, f be Function of X,Y such that
A1: g is continuous & f|X0 is continuous;
  (g*f)|X0 = g*(f|X0) by Th62;
  hence thesis by A1;
end;
