 reserve A, B, C, A1, B1, C1, A2, B2, C2, C3 for Point of TOP-REAL 2,
  lambda, mu, nu, alpha, beta, gamma for Real,
  X, Y, Z for Subset of TOP-REAL 2;

theorem
  Th10: the_area_of_polygon3(0. (TOP-REAL 2), B, C) =
  (B `1 * C `2 - C `1 * B `2) / 2
proof
  the_area_of_polygon3(0. (TOP-REAL 2), B, C) = ((0 * B `2 - B `1 *
  (|[0,0]|) `2) + (B `1 * C `2 - C `1 * B `2) + (C `1 * (|[0,0]|) `2 -
  (|[0,0]|) `1 * C `2)) / 2 by EUCLID:52,54
  .= ((0 - B `1 * 0) + (B `1 * C `2 - C `1 * B `2) + (C `1 * (|[0,0]|) `2 -
  (|[0,0]|) `1 * C `2)) / 2 by EUCLID:52
  .= ((B `1 * C `2 - C `1 * B `2) + (C `1 * 0 - (|[0,0]|) `1 * C `2)) / 2
  by EUCLID:52
  .= ((B `1 * C `2 - C `1 * B `2) + (0 - 0 * C `2)) / 2 by EUCLID:52;
  hence thesis;
end;
