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 Th82:
  for g being continuous Function of X0,Y holds X1 is SubSpace of
  X0 implies g|X1 is continuous Function of X1,Y
proof
  let g be continuous Function of X0,Y;
  assume
A1: X1 is SubSpace of X0;
  for x1 being Point of X1 holds g|X1 is_continuous_at x1
  proof
    let x1 be Point of X1;
    consider x0 being Point of X0 such that
A2: x0 = x1 by A1,Th10;
    g is_continuous_at x0 by Th44;
    hence thesis by A1,A2,Th74;
  end;
  hence thesis by Th44;
end;
