 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
  the_area_of_polygon3(A + A1, B, C) = the_area_of_polygon3(A, B, C) +
  the_area_of_polygon3(A1, B, C) - the_area_of_polygon3(0. (TOP-REAL 2), B, C)
proof
  the_area_of_polygon3(A + A1, B, C) = (((A `1 + A1 `1) * B `2 - B `1 * (A +
  A1) `2) + (B `1 * C `2 - C `1 * B `2) + (C `1 * (A + A1) `2 - (A + A1) `1 *
  C `2)) / 2 by Th1
  .= (((A `1 + A1 `1) * B `2 - B `1 * (A `2 + A1 `2)) + (B `1 * C `2 - C `1 *
  B `2) + (C `1 * (A + A1) `2 - (A + A1) `1 * C `2)) / 2 by Th1
  .= (((A `1 + A1 `1) * B `2 - B `1 * (A `2 + A1 `2)) + (B `1 * C `2 - C `1 *
  B `2) + (C `1 * (A `2 + A1 `2) - (A + A1) `1 * C `2)) / 2 by Th1
  .= (((A `1 + A1 `1) * B `2 - B `1 * (A `2 + A1 `2)) + (B `1 * C `2 - C `1 *
  B `2) + (C `1 * (A `2 + A1 `2) - (A `1 + A1 `1) * C `2)) / 2 by Th1
  .= (((A `1 + A1 `1) * B `2 - B `1 * (A `2 + A1 `2)) + 2 * (B `1 * C `2 -
  C `1 * B `2) + (C `1 * (A `2 + A1 `2) - (A `1 + A1 `1) * C `2)) / 2 - (B `1
  * C `2 - C `1 * B `2) / 2;
  hence thesis by Th10;
end;
