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;

theorem
  X1 is open SubSpace of X & X1 is SubSpace of X0 implies for x0 being
Point of X0, x1 being Point of X1 st x0 = x1 holds g is_continuous_at x0 iff g|
  X1 is_continuous_at x1
proof
  assume
A1: X1 is open SubSpace of X;
  assume
A2: X1 is SubSpace of X0;
  let x0 be Point of X0, x1 be Point of X1;
  assume
A3: x0 = x1;
  hence g is_continuous_at x0 implies g|X1 is_continuous_at x1 by A2,Th74;
  thus g|X1 is_continuous_at x1 implies g is_continuous_at x0
  proof
    the carrier of X1 c= the carrier of X0 by A2,TSEP_1:4;
    then
A4: X1 is open SubSpace of X0 by A1,TSEP_1:19;
    assume g|X1 is_continuous_at x1;
    hence thesis by A3,A4,Th79;
  end;
end;
