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;

theorem
  for f being Function of X,Y, X1, X2 being non empty SubSpace of X for
x being Point of X1 union X2, x1 being Point of X1, x2 being Point of X2 st x =
x1 & x = x2 holds f|(X1 union X2) is_continuous_at x iff f|X1 is_continuous_at
  x1 & f|X2 is_continuous_at x2
proof
  let f be Function of X,Y, X1, X2 be non empty SubSpace of X;
A1: X1 is SubSpace of X1 union X2 & X2 is SubSpace of X1 union X2 by TSEP_1:22;
  let x be Point of X1 union X2, x1 be Point of X1, x2 be Point of X2 such
  that
A2: x = x1 & x = x2;
  thus f|(X1 union X2) is_continuous_at x implies f|X1 is_continuous_at x1 & f
  |X2 is_continuous_at x2 by A2,A1,Th75;
  thus f|X1 is_continuous_at x1 & f|X2 is_continuous_at x2 implies f|(X1 union
  X2) is_continuous_at x
  proof
    set g = f|(X1 union X2);
    assume
A3: f|X1 is_continuous_at x1 & f|X2 is_continuous_at x2;
    g|X1 = f|X1 & g|X2 = f|X2 by A1,Th70;
    hence thesis by A2,A3,Th111;
  end;
end;
