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
  for X0 being open non empty SubSpace of X, x being Point of X, x0
  being Point of X0 st x = x0 holds f is_continuous_at x iff f|X0
  is_continuous_at x0
proof
  let X0 be open non empty SubSpace of X, x be Point of X, x0 be Point of X0;
  assume
A1: x = x0;
  hence f is_continuous_at x implies f|X0 is_continuous_at x0 by Th58;
  thus f|X0 is_continuous_at x0 implies f is_continuous_at x
  proof
    reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
    assume
A2: f|X0 is_continuous_at x0;
    A is open by TSEP_1:16;
    hence thesis by A1,A2,Th60;
  end;
end;
