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 SubSpace of X0 implies for A being Subset of X, x0 being Point
  of X0, x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 &
  x0 = x1 holds g is_continuous_at x0 iff g|X1 is_continuous_at x1
proof
  assume
A1: X1 is SubSpace of X0;
  let A be Subset of X, x0 be Point of X0, x1 be Point of X1 such that
A2: A is open and
A3: x0 in A and
A4: A c= the carrier of X1 and
A5: x0 = x1;
  thus g is_continuous_at x0 implies g|X1 is_continuous_at x1 by A1,A5,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 A1,TSEP_1:4;
    then reconsider B = A as Subset of X0 by A4,XBOOLE_1:1;
    assume
A6: g|X1 is_continuous_at x1;
    B is open by A2,TOPS_2:25;
    hence thesis by A1,A3,A4,A5,A6,Th77;
  end;
end;
